https://wiki.math.wisc.edu/api.php?action=feedcontributions&user=Klagsbru&feedformat=atomUW-Math Wiki - User contributions [en]2024-03-29T08:31:43ZUser contributionsMediaWiki 1.39.5https://wiki.math.wisc.edu/index.php?title=NTSGrad_Spring_2013&diff=5095NTSGrad Spring 20132013-02-26T18:56:32Z<p>Klagsbru: </p>
<hr />
<div>= Number Theory – Representation Theory Graduate Student Seminar, University of Wisconsin–Madison =<br />
<br />
*'''When:''' Tuesdays, 2:30pm–3:30pm,<br />
*'''Where:''' Van Vleck B129<br />
<br />
The purpose of this seminar is to have a talk on each Tuesday by a graduate student to<br />
help orient ourselves for the [[NTS|Number Theory Seminar]] talk on the following Thursday.<br />
These talks should be aimed at beginning graduate students, and should try to <br />
explain some of the background, terminology, and ideas for the Thursday talk. <br />
<br />
== Spring 2013 Semester ==<br />
<br />
<center><br />
<br />
{| style="color:black; font-size:120%" border="0" cellpadding="14" cellspacing="5"<br />
|-<br />
| bgcolor="#D0D0D0" width="300" align="center"|'''Date'''<br />
| bgcolor="#F0A0A0" width="300" align="center"|'''Speaker'''<br />
| bgcolor="#BCD2EE" width="300" align="center"|'''Title (click to see abstract)'''<br />
|- <br />
| bgcolor="#E0E0E0"| Jan 22 (Tues.)<br />
| bgcolor="#F0B0B0"| Organizational meeting<br />
| bgcolor="#BCE2FE"| Meet in 9th floor lounge<br />
|- <br />
| bgcolor="#E0E0E0"| Jan 29 (Tues.)<br />
| bgcolor="#F0B0B0"| Silas Johnson<br />
| bgcolor="#BCE2FE"|[[NTS Spring 2013/Abstracts#September 18 | <font color="black"><em>Moved to Wednesday, 1:30pm</em></font>]]<br />
|- <br />
| bgcolor="#E0E0E0"| Feb 5 (Tues.)<br />
| bgcolor="#F0B0B0"| Vlad Matei<br />
| bgcolor="#BCE2FE"|[[NTS Spring 2013/Abstracts#September 25 | <font color="black"><em>tba</em></font>]]<br />
|- <br />
| bgcolor="#E0E0E0"| Feb 12 (Tues.)<br />
| bgcolor="#F0B0B0"| Luanlei Zhao<br />
| bgcolor="#BCE2FE"|[[NTS Spring 2013/Abstracts#October 2 | <font color="black"><em>tba</em></font>]]<br />
|- <br />
| bgcolor="#E0E0E0"| Feb 19 (Tues.)<br />
| bgcolor="#F0B0B0"| No talk<br />
| bgcolor="#BCE2FE"|[[NTS Spring 2013/Abstracts#October 9 | <font color="black"><em>No talk</em></font>]]<br />
|- <br />
| bgcolor="#E0E0E0"| Feb 26 (Tues.)<br />
| bgcolor="#F0B0B0"| <br />
| bgcolor="#BCE2FE"|[[NTS Spring 2013/Abstracts#October 16 | <font color="black"><em>tba</em></font>]]<br />
|- <br />
| bgcolor="#E0E0E0"| Mar 5 (Tues.)<br />
| bgcolor="#F0B0B0"| Peng Yu<br />
| bgcolor="#BCE2FE"|[[NTS Spring 2013/Abstracts#October 23 | <font color="black"><em>tba</em></font>]]<br />
|- <br />
| bgcolor="#E0E0E0"| Mar 12 (Tues.)<br />
| bgcolor="#F0B0B0"| who?<br />
| bgcolor="#BCE2FE"|[[NTS Spring 2013/Abstracts#October 30 | <font color="black"><em>tba</em></font>]]<br />
|- <br />
| bgcolor="#E0E0E0"| Mar 19 (Tues.)<br />
| bgcolor="#F0B0B0"| who?<br />
| bgcolor="#BCE2FE"|[[NTS Spring 2013/Abstracts#November 6 | <font color="black"><em>tba</em></font>]]<br />
|- <br />
| bgcolor="#E0E0E0"| Mar 26 (Tues.)<br />
| bgcolor="#F0B0B0"| No seminar<br>(Spring break!)<br />
| bgcolor="#BCE2FE"| Spring break!<br />
|-<br />
| bgcolor="#E0E0E0"| Apr 2 (Tues.)<br />
| bgcolor="#F0B0B0"| Rachel Davis<br />
| bgcolor="#BCE2FE"|[[NTS Spring 2013/Abstracts#November 20 | <font color="black"><em>tba</em></font>]]<br />
|-<br />
| bgcolor="#E0E0E0"| Apr 9 (Tues.)<br />
| bgcolor="#F0B0B0"| Lalit Jain<br />
| bgcolor="#BCE2FE"|[[NTS Spring 2013/Abstracts#November 27 | <font color="black"><em>tba</em></font>]]<br />
|- <br />
| bgcolor="#E0E0E0"| Apr 16 (Tues.)<br />
| bgcolor="#F0B0B0"| Evan Dummit<br />
| bgcolor="#BCE2FE"|[[NTS Spring 2013/Abstracts#December 4 | <font color="black"><em>tba</em></font>]]<br />
|- <br />
| bgcolor="#E0E0E0"| Apr 23 (Tues.)<br />
| bgcolor="#F0B0B0"| who?<br />
| bgcolor="#BCE2FE"|[[NTS Spring 2013/Abstracts#December 4 | <font color="black"><em>tba</em></font>]]<br />
|- <br />
| bgcolor="#E0E0E0"| Apr 30 (Tues.)<br />
| bgcolor="#F0B0B0"| Megan Maguire<br />
| bgcolor="#BCE2FE"|[[NTS Spring 2013/Abstracts#December 4 | <font color="black"><em>tba</em></font>]]<br />
|- <br />
| bgcolor="#E0E0E0"| May 7 (Tues.)<br />
| bgcolor="#F0B0B0"| Yueke Hu<br />
| bgcolor="#BCE2FE"|[[NTS Spring 2013/Abstracts#December 4 | <font color="black"><em>tba</em></font>]]<br />
|}<br />
<br />
</center><br />
<br />
<br><br />
<br />
== Organizers ==<br />
<br />
[http://www.math.wisc.edu/~rharron/ Robert Harron]<br />
<br />
[http://www.math.wisc.edu/~klagsbru/ Zev Klagsbrun]<br />
<br />
Sean Rostami<br />
<br />
----<br />
The Fall 2012 NTS Grad page can be found [[NTSGrad Fall 2012|here]].<br />
----<br />
Return to the [[NTS|Number Theory Seminar Page]]<br />
<br />
Return to the [[Algebra|Algebra Group Page]]</div>Klagsbruhttps://wiki.math.wisc.edu/index.php?title=NTS_Spring_2012/Abstracts&diff=3765NTS Spring 2012/Abstracts2012-04-19T16:40:36Z<p>Klagsbru: /* May 3 */</p>
<hr />
<div>== February 2 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Evan Dummit''' (Madison)<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Kakeya sets over non-archimedean local rings<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: In a forthcoming paper with Marci Habliscek, we constructed a Kakeya set over the formal power series ring '''F'''<sub>''q''</sub><nowiki>[[</nowiki>''t''&thinsp;<nowiki>]]</nowiki>, answering a question posed by Ellenberg, Oberlin, and Tao. My talk will be devoted to explaining some of the older history of the Kakeya problem in analysis and the newer history of the Kakeya problem in combinatorics, including my joint work with Marci. In particular, I will give Dvir's solution of the Kakeya problem over finite fields, and explain the problem's extension to other classes of rings. <br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== February 16 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Tonghai Yang''' (Madison)<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: A little linear algebra on CM abelian surfaces<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: In this talk, I will discuss an interesting Hermitian form structure on the space of ''special endormorphisms'' of a CM abelian surface, and how to use it make a moduli problem and prove an arithmetic Siegel–Weil formula over a real quadratic field. This is a joint work with Ben Howard.<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== February 23 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Christelle Vincent''' (Madison)<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Drinfeld modular forms<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: We will begin by introducing the Drinfeld setting, and in particular Drinfeld modular forms and their connection to the geometry of Drinfeld modular curves. We will then present some results about Drinfeld modular forms that we obtained in the process of computing certain geometric points on Drinfeld modular curves. More precisely, we will talk about Drinfeld modular forms modulo ''P'', for ''P'' a prime ideal in '''F'''<sub>''q''</sub>[''T''&thinsp;], and about Drinfeld quasi-modular forms.<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== March 1 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Shamgar Gurevich''' (Madison)<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Computing the Matched Filter in Linear Time<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: In the digital radar problem we design a function (waveform) S(t) in the Hilbert space H&nbsp;=&nbsp;C('''Z'''/''p'') of complex valued functions on '''Z'''/''p''&nbsp;=&nbsp;{0,&nbsp;...,&nbsp;''p''&nbsp;&minus;&nbsp;1}, the integers modulo a prime number ''p''&nbsp;≫&nbsp;0. We transmit the function S(t) using the radar to the object that we want to detect. The wave S(t) hits the object, and is reflected back via the echo wave R(t) in H, which has the form <br />
<br />
R(t)&nbsp;=&nbsp;exp{2πiωt/''p''}⋅S(t+τ)&nbsp;+&nbsp;W(t), <br />
<br />
where W(t) in H is a white noise, and τ,&nbsp;ω in '''Z'''/''p'', encode the distance from, and velocity of, the object. <br />
<br />
Problem (digital radar problem) Extract τ,&nbsp;ω from R and S. <br />
<br />
In the lecture I first introduce the classical matched filter (MF) algorithm that suggests the 'traditional' way (using fast Fourier transform) to solve the digital radar problem in order of ''p''<sup>2</sup>⋅log(''p'') operations. I will then explain how to use techniques from group representation theory and arithmetic to design (construct) waveforms S(t) which enable us to introduce a fast matched filter (FMF) algorithm, that we call the "flag algorithm", which solves the digital radar problem in a much faster way of order of ''p''⋅log(''p'') operations. <br />
<br />
I will demonstrate additional applications to mobile communication, and global positioning system (GPS). <br />
<br />
This is a joint work with A. Fish (Math, Madison), R. Hadani (Math, Austin), A. Sayeed (Electrical Engineering, Madison), and O. Schwartz (Electrical Engineering and Computer Science, Berkeley). <br />
<br />
The lecture is suitable for general math/engineering audience.<br />
<br />
|} <br />
</center><br />
<br />
<br><br />
== March 8 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Zev Klagsbrun''' (Madison)<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Erdős–Kac Type Theorems<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: In its most popular formulation, the Erdős–Kac Theorem gives a distribution on the number of distinct primes factors (ω(''n'')) of the numbers up to ''N''. Variants of the Erdős–Kac Theorem yield distributions on additive functions in a surprising number of settings. This talk will outline the basics of the theory by focusing on some results of Granville and Soundararajan that allow one to easily prove Erdős–Kac type results for a variety of problems as well as present a recent result of my own using the Granville and Soundararajan framework.<br />
<br />
The lecture is suitable for general math audience.<br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== March 15 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Yongqiang Zhao''' (Madison)<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: On the Roberts conjecture<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: Let ''N''(''X'')&nbsp;=&nbsp;#{''K''&nbsp;<nowiki>|</nowiki>&nbsp;[''K'':'''Q''']&nbsp;=&nbsp;3,&nbsp;disc(''K'')&nbsp;≤&nbsp;''X''} be the counting function of cubic fields of bounded<br />
discriminant. The Roberts Conjecture is about the second term of this counting function. This conjecture was thought to be hard and dormant for some time. However, recently, four very different<br />
approaches to this problem were developed independently by Bhargava, Shankar and Tsimerman,<br />
Hough, Taniguchi and Thorne, and myself.<br />
In this talk, I will mention the first three approaches very briefly, then focus on my results on the function field case. I will give an outline of the proof. If time permits, I will indicate how the geometry<br />
feeds back to the number field case, in particular, how one could possibly define a new invariant<br />
for cubic fields.<br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== March 22 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Paul Terwilliger''' (Madison)<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Introduction to tridiagonal pairs<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: In this talk we consider a linear algebraic object called a tridiagonal pair. This object originated in algebraic graph theory, and has connections to orthogonal polynomials and representation theory. In our discussion we aim at a general mathematical audience; no prior experience with the above topics is assumed. Our main results are joint work with Tatsuro Ito and Kazumasa Nomura.<br />
<br />
The concept of a tridiagonal pair is best explained by starting with a special case called a Leonard pair. Let '''F''' denote a field, and let ''V'' denote a vector space over '''F''' with finite positive dimension. By a Leonard pair on ''V'' we mean a pair of linear transformations ''A''&nbsp;:&nbsp;''V''&nbsp;→&nbsp;''V'' and ''A''<sup>∗</sup>&nbsp;:&nbsp;''V''&nbsp;→&nbsp;''V'' that satisfy the following two conditions:<br />
#There exists a basis for ''V'' with respect to which the matrix representing ''A'' is diagonal and the matrix representing ''A''<sup>∗</sup> is irreducible tridiagonal;<br />
#There exists a basis for ''V'' with respect to which the matrix representing ''A''<sup>∗</sup> is diagonal and the matrix representing ''A'' is irreducible tridiagonal.<br />
<br />
We recall that a tridiagonal matrix is irreducible whenever each entry on the superdiagonal is nonzero and each entry on the subdiagonal is nonzero. The name "Leonard pair" is motivated by a connection to a 1982 theorem of the combinatorialist Doug Leonard involving the ''q''-Racah and related polynomials of the Askey scheme. Leonard’s theorem was heavily influenced by the work of Eiichi Bannai and Tatsuro Ito on ''P''- and ''Q''- polynomial association schemes, and the work of Richard Askey on orthogonal polynomials. These works in turn were influenced by the work of Philippe Delsarte on coding theory, dating from around 1973.<br />
<br />
The central result about Leonard pairs is that they are in bijection with the orthogonal polynomials that make up the terminating branch of the Askey scheme. This branch consists of the ''q''-Racah, ''q''-Hahn, dual ''q''-Hahn, ''q''-Krawtchouk, dual ''q''-Krawtchouk, quantum ''q''-Krawtchouk, affine ''q''-Krawtchouk, Racah, Hahn, dual-Hahn, Krawtchouk, and the Bannai/Ito polynomials. The bijection makes it possible to develop a uniform theory of these polynomials starting from the Leonard pair axiom, and we have done this over the past several years.<br />
<br />
A tridiagonal pair is a generalization of a Leonard pair and defined as follows. Let ''V'' denote a vector space over '''F''' with finite positive dimension. A ''tridiagonal pair on V'' is a pair of linear transformations ''A''&nbsp;:&nbsp;''V''&nbsp;→&nbsp;''V'' and ''A''<sup>∗</sup>&nbsp;:&nbsp;''V''&nbsp;→&nbsp;''V'' that satisfy the following four conditions:<br />
<ol><br />
<li>Each of ''A'', ''A''<sup>∗</sup> is diagonalizable on ''V'';<br />
<li>There exists an ordering {''V<sub>i</sub>''}<sub>''i''=0,...,''d''</sub> of the eigenspaces of ''A'' such that<br />
::''A''<sup>*</sup>''V<sub>i</sub>''&nbsp;⊆&nbsp;''V''<sub>''i''&minus;1</sub>&nbsp;+&nbsp;''V<sub>i</sub>''&nbsp;+&nbsp;''V''<sub>''i''+1</sub>&nbsp;&nbsp;(0&nbsp;≤&nbsp;''i''&nbsp;≤&nbsp;''d''),<br />
where ''V''<sub>&minus;1</sub>&nbsp;=&nbsp;0, ''V''<sub>''d''+1</sub>&nbsp;=&nbsp;0;<br />
<li>There exists an ordering {''V<sub>i</sub>''<sup>*</sup>}<sub>''i''=0,...,&delta;</sub> of the eigenspaces of ''A''<sup>*</sup> such that<br />
::''AV<sub>i</sub>''<sup>*</sup>&nbsp;⊆&nbsp;''V''<sup>*</sup><sub>''i''&minus;1</sub>&nbsp;+&nbsp;''V''<sup>*</sup><sub>''i''</sub>&nbsp;+&nbsp;''V''<sup>*</sup><sub>''i''+1</sub>&nbsp;&nbsp;(0&nbsp;≤&nbsp;''i''&nbsp;≤&nbsp;&delta;),<br />
where ''V''<sup>*</sup><sub>&minus;1</sub>&nbsp;=&nbsp;0, ''V''<sup>*</sup><sub>''d''+1</sub>&nbsp;=&nbsp;0;<br />
<li>There is no subspace ''W''&nbsp;⊆&nbsp;''V'' such that ''AW''&nbsp;⊆&nbsp;W, ''A''<sup>*</sup>''W''&nbsp;⊆&nbsp;''W'', ''W''&nbsp;&ne;&nbsp;0, ''W''&nbsp;&ne;&nbsp;''V''.<br />
</ol><br />
It turns out that ''d''&nbsp;=&nbsp;δ and this common value is called the diameter of the pair.<br />
A Leonard pair is the same thing as a tridiagonal pair for which the eigenspaces<br />
''V'' and ''V''<sup>∗</sup> all have dimension 1.<br />
<br />
Tridiagonal pairs arise naturally in the theory of ''P''- and ''Q''-polynomial association schemes, in connection with irreducible modules for the subconstituent algebra. They also appear in recent work of Pascal Baseilhac on the Ising model and related structures in statistical mechanics.<br />
<br />
In this talk we will summarize the basic facts about a tridiagonal pair, describing<br />
features such as the eigenvalues, dual eigenvalues, shape, tridiagonal relations,<br />
split decomposition, and parameter array. We will then focus on a special case<br />
said to be sharp and defined as follows. Referring to the tridiagonal pair ''A'', ''A''<sup>∗</sup> in the above definition, it turns out that for 0&nbsp;≤&nbsp;''i''&nbsp;≤&nbsp;''d'' the dimensions of ''V<sub>i</sub>'', ''V''<sub>''d''&minus;''i''</sub>, ''V''<sup>*</sup><sub>''i''</sub>, ''V''<sup>*</sup><sub>''d''&minus;''i''</sub> coincide; the pair ''A'', ''A''<sup>∗</sup> is called ''sharp'' whenever ''V''<sub>0</sub> has dimension 1. It is known that if '''F''' is algebraically closed then ''A'', ''A''<sup>∗</sup> is sharp.<br />
<br />
In our main result we classify the sharp tridiagonal pairs up to isomorphism.<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== March 29 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''David P. Roberts''' (U. Minnesota Morris)<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Lightly ramified number fields with Galois group ''S''.''M''<sub>12</sub>.''A''<br />
|-<br />
| bgcolor="#BCD2EE" |<br />
Abstract: Two of the most important invariants of an irreducible polynomial ''f''(''x'')&nbsp;&isin;&nbsp;'''Z'''[''x''&thinsp;] are its<br />
Galois group ''G'' and its field discriminant ''D''. The inverse Galois problem asks <br />
one to find a polynomial ''f''(''x'') having any prescribed Galois group ''G''. Refinements<br />
of this problem ask for ''D'' to be small in various senses, for example of the form<br />
&plusmn;&thinsp;''p<sup>a''</sup> for the smallest possible prime ''p''. <br />
<br />
The talk will discuss this problem in general, with a focus on the technique of <br />
specializing three-point covers for solving instances of it. Then it will pursue the cases of the <br />
Mathieu group ''M''<sub>12</sub>, its automorphism group ''M''<sub>12</sub>.2, its double cover<br />
2.''M''<sub>12</sub>, and the combined extension 2.''M''<sub>12</sub>.2. Among the polynomials<br />
found is<br />
{| style="background: #BCD2EE;" align="center"<br />
|-<br />
| ''f''(''x'')&nbsp;= || ''x''<sup>48</sup> + 2 ''e''<sup>3</sup> ''x''<sup>42</sup> + 69 ''e''<sup>5</sup> ''x''<sup>36</sup> + 868 ''e''<sup>7</sup> ''x''<sup>30</sup> &minus; 4174 ''e''<sup>7</sup> ''x''<sup>26</sup> + 11287 ''e''<sup>9</sup> ''x''<sup>24</sup><br />
|-<br />
| ||&minus; 4174 ''e''<sup>10</sup> ''x''<sup>20</sup> + 5340 ''e''<sup>12</sup> ''x''<sup>18</sup> + 131481 ''e''<sup>12</sup> ''x''<sup>14</sup> +17599 ''e''<sup>14</sup> ''x''<sup>12</sup> + 530098 ''e''<sup>14</sup> ''x''<sup>8</sup><br />
|-<br />
| ||+ 3910 ''e''<sup>16</sup> ''x''<sup>6</sup> + 4355569 ''e''<sup>14</sup> ''x''<sup>4</sup> + 20870 ''e''<sup>16</sup> ''x''<sup>2</sup> + 729 ''e''<sup>18</sup>,<br />
|}<br />
<br />
with ''e'' = 11. This polynomial has Galois group ''G''&nbsp;=&nbsp;2.''M''<sub>12</sub>.2 and<br />
field discriminant 11<sup>88</sup>. It makes ''M''<sub>12</sub> the<br />
first of the twenty-six sporadic simple groups &Gamma;<br />
known to have a polynomial with Galois group <br />
''G'' involving &Gamma; and field discriminant ''D'' <br />
the power of a single prime dividing |&Gamma;&thinsp;|.<br />
<br />
|} <br />
</center><br />
<br />
<br />
<br><br />
<br />
== April 12 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Chenyan Wu''' (Minnesota)<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Rallis inner product formula for theta lifts from metaplectic groups to orthogonal groups<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: Let π be a genuine cuspidal representation of the metaplectic group of rank<br />
''n''. We consider the theta lifts to the orthogonal group associated to a<br />
quadratic space of dimension 2''n''&nbsp;+&nbsp;1. We show a case of a regularized Rallis inner<br />
product formula that relates the pairing of theta lifts to the central value of the<br />
Langlands ''L''-function of π twisted by a genuine character. This enables us to<br />
demonstrate the relation between non-vanishing of theta lifts and the non-vanishing of<br />
central ''L''-values. We prove also a case of regularized Siegel–Weil formula which is<br />
missing in the literature, as it forms the basis of our proof of the Rallis inner<br />
product formula.<br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
<br />
==<span id="April 16"></span> April 16 (special day: '''Monday''', special time: '''3:30pm–4:30pm''', special place: '''VV B139''') ==<br />
<br />
<center><br />
<br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Hourong Qin''' (Nanjing U., China)<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: CM elliptic curves and quadratic polynomials representing primes <br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: No polynomial of degree two or higher has been proved to represent infinitely many primes. Let ''E'' be an elliptic curve defined over '''Q''' with complex multiplication. Fix an integer ''r''. We give sufficient and necessary conditions for ''a<sub>p''</sub>&nbsp;=&nbsp;''r'' for some prime ''p''. We show that there are infinitely many primes ''p'' such that ''a<sub>p''</sub>&nbsp;=&nbsp;''r'' for some fixed integer ''r'' if<br />
and only if a quadratic polynomial represents infinitely many primes ''p''.<br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== April 19 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Robert Guralnick''' (U. Southern California)<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: A variant of Burnside and Galois representations which are automorphic<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: Wiles, Taylor, Harris and others used the notion of a big<br />
representation of a finite<br />
group to show that certain representations are automorphic. Jack Thorne<br />
recently observed<br />
that one could weaken this notion of bigness to get the same conclusions. He<br />
called this property adequate. An absolutely irreducible representation ''V''<br />
of a finite group ''G'' in characteristic ''p'' is called adequate if ''G'' has<br />
no ''p''-quotients, the dimension<br />
of ''V'' is prime to ''p'', ''V'' has non-trivial self extensions and End(''V'') is<br />
generated by the linear<br />
span of the elements of order prime to ''p'' in ''G''. If ''G'' has order<br />
prime to ''p'', all of these conditions<br />
hold&mdash;the last condition is sometimes called Burnside's Lemma. We<br />
will discuss a recent<br />
result of Guralnick, Herzig, Taylor and Thorne showing that if ''p'' ><br />
2 dim ''V'' + 2, then<br />
any absolutely irreducible representation is adequate. We will also<br />
discuss some examples<br />
showing that the span of the ''p'''-elements in End(''V'') need not be all of End(''V'').<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== April 26 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Frank Thorne''' (U. South Carolina)<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Secondary terms in counting functions for cubic fields<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
<br />
Abstract: We will discuss our proof of secondary terms of order ''X''<sup>5/6</sup> in the Davenport–Heilbronn<br />
theorems on cubic fields and 3-torsion in class groups of quadratic fields. For cubic<br />
fields this confirms a conjecture of Datskovsky–Wright and Roberts. We also will describe<br />
some generalizations, in particular to arithmetic progressions, where we discover a<br />
curious bias in the secondary term.<br />
<br />
Roberts’ conjecture has also been proved independently by Bhargava, Shankar, and<br />
Tsimerman. Their proof uses the geometry of numbers, while our proof uses the analytic<br />
theory of Shintani zeta functions.<br />
<br />
We will also discuss a combined approach which yields further improved error terms. If<br />
there is time (or after the talk), I will also discuss a couple of side projects and my<br />
plans for further related work.<br />
<br />
This is joint work with Takashi Taniguchi.<br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== May 3 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Alina Cojocaru''' (U. Illinois at Chicago)<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Frobenius fields for elliptic curves<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: Let ''E'' be an elliptic curve defined over '''Q'''. For a prime p of good reduction for ''E'', let &pi;<sub>p</sub> be the p-Weil root of E and '''Q'''(&pi;<sub>p</sub>) the associated imaginary quadratic field generated by &pi;<sub>p</sub>. In 1976, Serge Lang and Hale Trotter formulated a conjectural asymptotic formula for the number of primes ''p < x'' for which '''Q'''(&pi;<sub>p</sub>) is isomorphic to a fixed imaginary quadratic field. I will discuss progress on this conjecture, in particular an average result confirming the predicted asymptotic formula. This is joint work with Henryk Iwaniec and Nathan Jones. <br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== May 10 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Samit Dasgupta''' (UC Santa Cruz)<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: tba<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: tba <br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== Organizer contact information ==<br />
<br />
[http://www.math.wisc.edu/~shamgar/ Shamgar Gurevich]<br />
<br />
[http://www.math.wisc.edu/~rharron/ Robert Harron]<br />
<br />
[http://www.math.wisc.edu/~klagsbru/ Zev Klagsbrun]<br />
<br />
[http://www.math.wisc.edu/~mmwood/ Melanie Matchett Wood]<br />
<br><br />
<br />
<br />
<br />
----<br />
Return to the [[NTS|Number Theory Seminar Page]]<br />
<br />
Return to the [[Algebra|Algebra Group Page]]</div>Klagsbruhttps://wiki.math.wisc.edu/index.php?title=NTS_Spring_2012/Abstracts&diff=3764NTS Spring 2012/Abstracts2012-04-19T16:40:24Z<p>Klagsbru: /* May 3 */</p>
<hr />
<div>== February 2 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Evan Dummit''' (Madison)<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Kakeya sets over non-archimedean local rings<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: In a forthcoming paper with Marci Habliscek, we constructed a Kakeya set over the formal power series ring '''F'''<sub>''q''</sub><nowiki>[[</nowiki>''t''&thinsp;<nowiki>]]</nowiki>, answering a question posed by Ellenberg, Oberlin, and Tao. My talk will be devoted to explaining some of the older history of the Kakeya problem in analysis and the newer history of the Kakeya problem in combinatorics, including my joint work with Marci. In particular, I will give Dvir's solution of the Kakeya problem over finite fields, and explain the problem's extension to other classes of rings. <br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== February 16 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Tonghai Yang''' (Madison)<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: A little linear algebra on CM abelian surfaces<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: In this talk, I will discuss an interesting Hermitian form structure on the space of ''special endormorphisms'' of a CM abelian surface, and how to use it make a moduli problem and prove an arithmetic Siegel–Weil formula over a real quadratic field. This is a joint work with Ben Howard.<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== February 23 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Christelle Vincent''' (Madison)<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Drinfeld modular forms<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: We will begin by introducing the Drinfeld setting, and in particular Drinfeld modular forms and their connection to the geometry of Drinfeld modular curves. We will then present some results about Drinfeld modular forms that we obtained in the process of computing certain geometric points on Drinfeld modular curves. More precisely, we will talk about Drinfeld modular forms modulo ''P'', for ''P'' a prime ideal in '''F'''<sub>''q''</sub>[''T''&thinsp;], and about Drinfeld quasi-modular forms.<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== March 1 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Shamgar Gurevich''' (Madison)<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Computing the Matched Filter in Linear Time<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: In the digital radar problem we design a function (waveform) S(t) in the Hilbert space H&nbsp;=&nbsp;C('''Z'''/''p'') of complex valued functions on '''Z'''/''p''&nbsp;=&nbsp;{0,&nbsp;...,&nbsp;''p''&nbsp;&minus;&nbsp;1}, the integers modulo a prime number ''p''&nbsp;≫&nbsp;0. We transmit the function S(t) using the radar to the object that we want to detect. The wave S(t) hits the object, and is reflected back via the echo wave R(t) in H, which has the form <br />
<br />
R(t)&nbsp;=&nbsp;exp{2πiωt/''p''}⋅S(t+τ)&nbsp;+&nbsp;W(t), <br />
<br />
where W(t) in H is a white noise, and τ,&nbsp;ω in '''Z'''/''p'', encode the distance from, and velocity of, the object. <br />
<br />
Problem (digital radar problem) Extract τ,&nbsp;ω from R and S. <br />
<br />
In the lecture I first introduce the classical matched filter (MF) algorithm that suggests the 'traditional' way (using fast Fourier transform) to solve the digital radar problem in order of ''p''<sup>2</sup>⋅log(''p'') operations. I will then explain how to use techniques from group representation theory and arithmetic to design (construct) waveforms S(t) which enable us to introduce a fast matched filter (FMF) algorithm, that we call the "flag algorithm", which solves the digital radar problem in a much faster way of order of ''p''⋅log(''p'') operations. <br />
<br />
I will demonstrate additional applications to mobile communication, and global positioning system (GPS). <br />
<br />
This is a joint work with A. Fish (Math, Madison), R. Hadani (Math, Austin), A. Sayeed (Electrical Engineering, Madison), and O. Schwartz (Electrical Engineering and Computer Science, Berkeley). <br />
<br />
The lecture is suitable for general math/engineering audience.<br />
<br />
|} <br />
</center><br />
<br />
<br><br />
== March 8 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Zev Klagsbrun''' (Madison)<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Erdős–Kac Type Theorems<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: In its most popular formulation, the Erdős–Kac Theorem gives a distribution on the number of distinct primes factors (ω(''n'')) of the numbers up to ''N''. Variants of the Erdős–Kac Theorem yield distributions on additive functions in a surprising number of settings. This talk will outline the basics of the theory by focusing on some results of Granville and Soundararajan that allow one to easily prove Erdős–Kac type results for a variety of problems as well as present a recent result of my own using the Granville and Soundararajan framework.<br />
<br />
The lecture is suitable for general math audience.<br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== March 15 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Yongqiang Zhao''' (Madison)<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: On the Roberts conjecture<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: Let ''N''(''X'')&nbsp;=&nbsp;#{''K''&nbsp;<nowiki>|</nowiki>&nbsp;[''K'':'''Q''']&nbsp;=&nbsp;3,&nbsp;disc(''K'')&nbsp;≤&nbsp;''X''} be the counting function of cubic fields of bounded<br />
discriminant. The Roberts Conjecture is about the second term of this counting function. This conjecture was thought to be hard and dormant for some time. However, recently, four very different<br />
approaches to this problem were developed independently by Bhargava, Shankar and Tsimerman,<br />
Hough, Taniguchi and Thorne, and myself.<br />
In this talk, I will mention the first three approaches very briefly, then focus on my results on the function field case. I will give an outline of the proof. If time permits, I will indicate how the geometry<br />
feeds back to the number field case, in particular, how one could possibly define a new invariant<br />
for cubic fields.<br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== March 22 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Paul Terwilliger''' (Madison)<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Introduction to tridiagonal pairs<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: In this talk we consider a linear algebraic object called a tridiagonal pair. This object originated in algebraic graph theory, and has connections to orthogonal polynomials and representation theory. In our discussion we aim at a general mathematical audience; no prior experience with the above topics is assumed. Our main results are joint work with Tatsuro Ito and Kazumasa Nomura.<br />
<br />
The concept of a tridiagonal pair is best explained by starting with a special case called a Leonard pair. Let '''F''' denote a field, and let ''V'' denote a vector space over '''F''' with finite positive dimension. By a Leonard pair on ''V'' we mean a pair of linear transformations ''A''&nbsp;:&nbsp;''V''&nbsp;→&nbsp;''V'' and ''A''<sup>∗</sup>&nbsp;:&nbsp;''V''&nbsp;→&nbsp;''V'' that satisfy the following two conditions:<br />
#There exists a basis for ''V'' with respect to which the matrix representing ''A'' is diagonal and the matrix representing ''A''<sup>∗</sup> is irreducible tridiagonal;<br />
#There exists a basis for ''V'' with respect to which the matrix representing ''A''<sup>∗</sup> is diagonal and the matrix representing ''A'' is irreducible tridiagonal.<br />
<br />
We recall that a tridiagonal matrix is irreducible whenever each entry on the superdiagonal is nonzero and each entry on the subdiagonal is nonzero. The name "Leonard pair" is motivated by a connection to a 1982 theorem of the combinatorialist Doug Leonard involving the ''q''-Racah and related polynomials of the Askey scheme. Leonard’s theorem was heavily influenced by the work of Eiichi Bannai and Tatsuro Ito on ''P''- and ''Q''- polynomial association schemes, and the work of Richard Askey on orthogonal polynomials. These works in turn were influenced by the work of Philippe Delsarte on coding theory, dating from around 1973.<br />
<br />
The central result about Leonard pairs is that they are in bijection with the orthogonal polynomials that make up the terminating branch of the Askey scheme. This branch consists of the ''q''-Racah, ''q''-Hahn, dual ''q''-Hahn, ''q''-Krawtchouk, dual ''q''-Krawtchouk, quantum ''q''-Krawtchouk, affine ''q''-Krawtchouk, Racah, Hahn, dual-Hahn, Krawtchouk, and the Bannai/Ito polynomials. The bijection makes it possible to develop a uniform theory of these polynomials starting from the Leonard pair axiom, and we have done this over the past several years.<br />
<br />
A tridiagonal pair is a generalization of a Leonard pair and defined as follows. Let ''V'' denote a vector space over '''F''' with finite positive dimension. A ''tridiagonal pair on V'' is a pair of linear transformations ''A''&nbsp;:&nbsp;''V''&nbsp;→&nbsp;''V'' and ''A''<sup>∗</sup>&nbsp;:&nbsp;''V''&nbsp;→&nbsp;''V'' that satisfy the following four conditions:<br />
<ol><br />
<li>Each of ''A'', ''A''<sup>∗</sup> is diagonalizable on ''V'';<br />
<li>There exists an ordering {''V<sub>i</sub>''}<sub>''i''=0,...,''d''</sub> of the eigenspaces of ''A'' such that<br />
::''A''<sup>*</sup>''V<sub>i</sub>''&nbsp;⊆&nbsp;''V''<sub>''i''&minus;1</sub>&nbsp;+&nbsp;''V<sub>i</sub>''&nbsp;+&nbsp;''V''<sub>''i''+1</sub>&nbsp;&nbsp;(0&nbsp;≤&nbsp;''i''&nbsp;≤&nbsp;''d''),<br />
where ''V''<sub>&minus;1</sub>&nbsp;=&nbsp;0, ''V''<sub>''d''+1</sub>&nbsp;=&nbsp;0;<br />
<li>There exists an ordering {''V<sub>i</sub>''<sup>*</sup>}<sub>''i''=0,...,&delta;</sub> of the eigenspaces of ''A''<sup>*</sup> such that<br />
::''AV<sub>i</sub>''<sup>*</sup>&nbsp;⊆&nbsp;''V''<sup>*</sup><sub>''i''&minus;1</sub>&nbsp;+&nbsp;''V''<sup>*</sup><sub>''i''</sub>&nbsp;+&nbsp;''V''<sup>*</sup><sub>''i''+1</sub>&nbsp;&nbsp;(0&nbsp;≤&nbsp;''i''&nbsp;≤&nbsp;&delta;),<br />
where ''V''<sup>*</sup><sub>&minus;1</sub>&nbsp;=&nbsp;0, ''V''<sup>*</sup><sub>''d''+1</sub>&nbsp;=&nbsp;0;<br />
<li>There is no subspace ''W''&nbsp;⊆&nbsp;''V'' such that ''AW''&nbsp;⊆&nbsp;W, ''A''<sup>*</sup>''W''&nbsp;⊆&nbsp;''W'', ''W''&nbsp;&ne;&nbsp;0, ''W''&nbsp;&ne;&nbsp;''V''.<br />
</ol><br />
It turns out that ''d''&nbsp;=&nbsp;δ and this common value is called the diameter of the pair.<br />
A Leonard pair is the same thing as a tridiagonal pair for which the eigenspaces<br />
''V'' and ''V''<sup>∗</sup> all have dimension 1.<br />
<br />
Tridiagonal pairs arise naturally in the theory of ''P''- and ''Q''-polynomial association schemes, in connection with irreducible modules for the subconstituent algebra. They also appear in recent work of Pascal Baseilhac on the Ising model and related structures in statistical mechanics.<br />
<br />
In this talk we will summarize the basic facts about a tridiagonal pair, describing<br />
features such as the eigenvalues, dual eigenvalues, shape, tridiagonal relations,<br />
split decomposition, and parameter array. We will then focus on a special case<br />
said to be sharp and defined as follows. Referring to the tridiagonal pair ''A'', ''A''<sup>∗</sup> in the above definition, it turns out that for 0&nbsp;≤&nbsp;''i''&nbsp;≤&nbsp;''d'' the dimensions of ''V<sub>i</sub>'', ''V''<sub>''d''&minus;''i''</sub>, ''V''<sup>*</sup><sub>''i''</sub>, ''V''<sup>*</sup><sub>''d''&minus;''i''</sub> coincide; the pair ''A'', ''A''<sup>∗</sup> is called ''sharp'' whenever ''V''<sub>0</sub> has dimension 1. It is known that if '''F''' is algebraically closed then ''A'', ''A''<sup>∗</sup> is sharp.<br />
<br />
In our main result we classify the sharp tridiagonal pairs up to isomorphism.<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== March 29 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''David P. Roberts''' (U. Minnesota Morris)<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Lightly ramified number fields with Galois group ''S''.''M''<sub>12</sub>.''A''<br />
|-<br />
| bgcolor="#BCD2EE" |<br />
Abstract: Two of the most important invariants of an irreducible polynomial ''f''(''x'')&nbsp;&isin;&nbsp;'''Z'''[''x''&thinsp;] are its<br />
Galois group ''G'' and its field discriminant ''D''. The inverse Galois problem asks <br />
one to find a polynomial ''f''(''x'') having any prescribed Galois group ''G''. Refinements<br />
of this problem ask for ''D'' to be small in various senses, for example of the form<br />
&plusmn;&thinsp;''p<sup>a''</sup> for the smallest possible prime ''p''. <br />
<br />
The talk will discuss this problem in general, with a focus on the technique of <br />
specializing three-point covers for solving instances of it. Then it will pursue the cases of the <br />
Mathieu group ''M''<sub>12</sub>, its automorphism group ''M''<sub>12</sub>.2, its double cover<br />
2.''M''<sub>12</sub>, and the combined extension 2.''M''<sub>12</sub>.2. Among the polynomials<br />
found is<br />
{| style="background: #BCD2EE;" align="center"<br />
|-<br />
| ''f''(''x'')&nbsp;= || ''x''<sup>48</sup> + 2 ''e''<sup>3</sup> ''x''<sup>42</sup> + 69 ''e''<sup>5</sup> ''x''<sup>36</sup> + 868 ''e''<sup>7</sup> ''x''<sup>30</sup> &minus; 4174 ''e''<sup>7</sup> ''x''<sup>26</sup> + 11287 ''e''<sup>9</sup> ''x''<sup>24</sup><br />
|-<br />
| ||&minus; 4174 ''e''<sup>10</sup> ''x''<sup>20</sup> + 5340 ''e''<sup>12</sup> ''x''<sup>18</sup> + 131481 ''e''<sup>12</sup> ''x''<sup>14</sup> +17599 ''e''<sup>14</sup> ''x''<sup>12</sup> + 530098 ''e''<sup>14</sup> ''x''<sup>8</sup><br />
|-<br />
| ||+ 3910 ''e''<sup>16</sup> ''x''<sup>6</sup> + 4355569 ''e''<sup>14</sup> ''x''<sup>4</sup> + 20870 ''e''<sup>16</sup> ''x''<sup>2</sup> + 729 ''e''<sup>18</sup>,<br />
|}<br />
<br />
with ''e'' = 11. This polynomial has Galois group ''G''&nbsp;=&nbsp;2.''M''<sub>12</sub>.2 and<br />
field discriminant 11<sup>88</sup>. It makes ''M''<sub>12</sub> the<br />
first of the twenty-six sporadic simple groups &Gamma;<br />
known to have a polynomial with Galois group <br />
''G'' involving &Gamma; and field discriminant ''D'' <br />
the power of a single prime dividing |&Gamma;&thinsp;|.<br />
<br />
|} <br />
</center><br />
<br />
<br />
<br><br />
<br />
== April 12 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Chenyan Wu''' (Minnesota)<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Rallis inner product formula for theta lifts from metaplectic groups to orthogonal groups<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: Let π be a genuine cuspidal representation of the metaplectic group of rank<br />
''n''. We consider the theta lifts to the orthogonal group associated to a<br />
quadratic space of dimension 2''n''&nbsp;+&nbsp;1. We show a case of a regularized Rallis inner<br />
product formula that relates the pairing of theta lifts to the central value of the<br />
Langlands ''L''-function of π twisted by a genuine character. This enables us to<br />
demonstrate the relation between non-vanishing of theta lifts and the non-vanishing of<br />
central ''L''-values. We prove also a case of regularized Siegel–Weil formula which is<br />
missing in the literature, as it forms the basis of our proof of the Rallis inner<br />
product formula.<br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
<br />
==<span id="April 16"></span> April 16 (special day: '''Monday''', special time: '''3:30pm–4:30pm''', special place: '''VV B139''') ==<br />
<br />
<center><br />
<br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Hourong Qin''' (Nanjing U., China)<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: CM elliptic curves and quadratic polynomials representing primes <br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: No polynomial of degree two or higher has been proved to represent infinitely many primes. Let ''E'' be an elliptic curve defined over '''Q''' with complex multiplication. Fix an integer ''r''. We give sufficient and necessary conditions for ''a<sub>p''</sub>&nbsp;=&nbsp;''r'' for some prime ''p''. We show that there are infinitely many primes ''p'' such that ''a<sub>p''</sub>&nbsp;=&nbsp;''r'' for some fixed integer ''r'' if<br />
and only if a quadratic polynomial represents infinitely many primes ''p''.<br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== April 19 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Robert Guralnick''' (U. Southern California)<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: A variant of Burnside and Galois representations which are automorphic<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: Wiles, Taylor, Harris and others used the notion of a big<br />
representation of a finite<br />
group to show that certain representations are automorphic. Jack Thorne<br />
recently observed<br />
that one could weaken this notion of bigness to get the same conclusions. He<br />
called this property adequate. An absolutely irreducible representation ''V''<br />
of a finite group ''G'' in characteristic ''p'' is called adequate if ''G'' has<br />
no ''p''-quotients, the dimension<br />
of ''V'' is prime to ''p'', ''V'' has non-trivial self extensions and End(''V'') is<br />
generated by the linear<br />
span of the elements of order prime to ''p'' in ''G''. If ''G'' has order<br />
prime to ''p'', all of these conditions<br />
hold&mdash;the last condition is sometimes called Burnside's Lemma. We<br />
will discuss a recent<br />
result of Guralnick, Herzig, Taylor and Thorne showing that if ''p'' ><br />
2 dim ''V'' + 2, then<br />
any absolutely irreducible representation is adequate. We will also<br />
discuss some examples<br />
showing that the span of the ''p'''-elements in End(''V'') need not be all of End(''V'').<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== April 26 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Frank Thorne''' (U. South Carolina)<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Secondary terms in counting functions for cubic fields<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
<br />
Abstract: We will discuss our proof of secondary terms of order ''X''<sup>5/6</sup> in the Davenport–Heilbronn<br />
theorems on cubic fields and 3-torsion in class groups of quadratic fields. For cubic<br />
fields this confirms a conjecture of Datskovsky–Wright and Roberts. We also will describe<br />
some generalizations, in particular to arithmetic progressions, where we discover a<br />
curious bias in the secondary term.<br />
<br />
Roberts’ conjecture has also been proved independently by Bhargava, Shankar, and<br />
Tsimerman. Their proof uses the geometry of numbers, while our proof uses the analytic<br />
theory of Shintani zeta functions.<br />
<br />
We will also discuss a combined approach which yields further improved error terms. If<br />
there is time (or after the talk), I will also discuss a couple of side projects and my<br />
plans for further related work.<br />
<br />
This is joint work with Takashi Taniguchi.<br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== May 3 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Alina Cojocaru''' (U. Illinois at Chicago)<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Frobenius fields for elliptic curves<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: Let ''E'' be an elliptic curve defined over '''Q'''. For a prime p of good reduction for ''E'', let &pi;<sub>p</sub> be the p-Weil root of E and Q(&pi;<sub>p</sub>) the associated imaginary quadratic field generated by &pi;<sub>p</sub>. In 1976, Serge Lang and Hale Trotter formulated a conjectural asymptotic formula for the number of primes ''p < x'' for which '''Q'''(&pi;<sub>p</sub>) is isomorphic to a fixed imaginary quadratic field. I will discuss progress on this conjecture, in particular an average result confirming the predicted asymptotic formula. This is joint work with Henryk Iwaniec and Nathan Jones. <br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== May 10 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Samit Dasgupta''' (UC Santa Cruz)<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: tba<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: tba <br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== Organizer contact information ==<br />
<br />
[http://www.math.wisc.edu/~shamgar/ Shamgar Gurevich]<br />
<br />
[http://www.math.wisc.edu/~rharron/ Robert Harron]<br />
<br />
[http://www.math.wisc.edu/~klagsbru/ Zev Klagsbrun]<br />
<br />
[http://www.math.wisc.edu/~mmwood/ Melanie Matchett Wood]<br />
<br><br />
<br />
<br />
<br />
----<br />
Return to the [[NTS|Number Theory Seminar Page]]<br />
<br />
Return to the [[Algebra|Algebra Group Page]]</div>Klagsbruhttps://wiki.math.wisc.edu/index.php?title=NTS_Spring_2012/Abstracts&diff=3763NTS Spring 2012/Abstracts2012-04-19T16:40:01Z<p>Klagsbru: /* May 3 */</p>
<hr />
<div>== February 2 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Evan Dummit''' (Madison)<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Kakeya sets over non-archimedean local rings<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: In a forthcoming paper with Marci Habliscek, we constructed a Kakeya set over the formal power series ring '''F'''<sub>''q''</sub><nowiki>[[</nowiki>''t''&thinsp;<nowiki>]]</nowiki>, answering a question posed by Ellenberg, Oberlin, and Tao. My talk will be devoted to explaining some of the older history of the Kakeya problem in analysis and the newer history of the Kakeya problem in combinatorics, including my joint work with Marci. In particular, I will give Dvir's solution of the Kakeya problem over finite fields, and explain the problem's extension to other classes of rings. <br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== February 16 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Tonghai Yang''' (Madison)<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: A little linear algebra on CM abelian surfaces<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: In this talk, I will discuss an interesting Hermitian form structure on the space of ''special endormorphisms'' of a CM abelian surface, and how to use it make a moduli problem and prove an arithmetic Siegel–Weil formula over a real quadratic field. This is a joint work with Ben Howard.<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== February 23 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Christelle Vincent''' (Madison)<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Drinfeld modular forms<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: We will begin by introducing the Drinfeld setting, and in particular Drinfeld modular forms and their connection to the geometry of Drinfeld modular curves. We will then present some results about Drinfeld modular forms that we obtained in the process of computing certain geometric points on Drinfeld modular curves. More precisely, we will talk about Drinfeld modular forms modulo ''P'', for ''P'' a prime ideal in '''F'''<sub>''q''</sub>[''T''&thinsp;], and about Drinfeld quasi-modular forms.<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== March 1 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Shamgar Gurevich''' (Madison)<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Computing the Matched Filter in Linear Time<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: In the digital radar problem we design a function (waveform) S(t) in the Hilbert space H&nbsp;=&nbsp;C('''Z'''/''p'') of complex valued functions on '''Z'''/''p''&nbsp;=&nbsp;{0,&nbsp;...,&nbsp;''p''&nbsp;&minus;&nbsp;1}, the integers modulo a prime number ''p''&nbsp;≫&nbsp;0. We transmit the function S(t) using the radar to the object that we want to detect. The wave S(t) hits the object, and is reflected back via the echo wave R(t) in H, which has the form <br />
<br />
R(t)&nbsp;=&nbsp;exp{2πiωt/''p''}⋅S(t+τ)&nbsp;+&nbsp;W(t), <br />
<br />
where W(t) in H is a white noise, and τ,&nbsp;ω in '''Z'''/''p'', encode the distance from, and velocity of, the object. <br />
<br />
Problem (digital radar problem) Extract τ,&nbsp;ω from R and S. <br />
<br />
In the lecture I first introduce the classical matched filter (MF) algorithm that suggests the 'traditional' way (using fast Fourier transform) to solve the digital radar problem in order of ''p''<sup>2</sup>⋅log(''p'') operations. I will then explain how to use techniques from group representation theory and arithmetic to design (construct) waveforms S(t) which enable us to introduce a fast matched filter (FMF) algorithm, that we call the "flag algorithm", which solves the digital radar problem in a much faster way of order of ''p''⋅log(''p'') operations. <br />
<br />
I will demonstrate additional applications to mobile communication, and global positioning system (GPS). <br />
<br />
This is a joint work with A. Fish (Math, Madison), R. Hadani (Math, Austin), A. Sayeed (Electrical Engineering, Madison), and O. Schwartz (Electrical Engineering and Computer Science, Berkeley). <br />
<br />
The lecture is suitable for general math/engineering audience.<br />
<br />
|} <br />
</center><br />
<br />
<br><br />
== March 8 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Zev Klagsbrun''' (Madison)<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Erdős–Kac Type Theorems<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: In its most popular formulation, the Erdős–Kac Theorem gives a distribution on the number of distinct primes factors (ω(''n'')) of the numbers up to ''N''. Variants of the Erdős–Kac Theorem yield distributions on additive functions in a surprising number of settings. This talk will outline the basics of the theory by focusing on some results of Granville and Soundararajan that allow one to easily prove Erdős–Kac type results for a variety of problems as well as present a recent result of my own using the Granville and Soundararajan framework.<br />
<br />
The lecture is suitable for general math audience.<br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== March 15 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Yongqiang Zhao''' (Madison)<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: On the Roberts conjecture<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: Let ''N''(''X'')&nbsp;=&nbsp;#{''K''&nbsp;<nowiki>|</nowiki>&nbsp;[''K'':'''Q''']&nbsp;=&nbsp;3,&nbsp;disc(''K'')&nbsp;≤&nbsp;''X''} be the counting function of cubic fields of bounded<br />
discriminant. The Roberts Conjecture is about the second term of this counting function. This conjecture was thought to be hard and dormant for some time. However, recently, four very different<br />
approaches to this problem were developed independently by Bhargava, Shankar and Tsimerman,<br />
Hough, Taniguchi and Thorne, and myself.<br />
In this talk, I will mention the first three approaches very briefly, then focus on my results on the function field case. I will give an outline of the proof. If time permits, I will indicate how the geometry<br />
feeds back to the number field case, in particular, how one could possibly define a new invariant<br />
for cubic fields.<br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== March 22 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Paul Terwilliger''' (Madison)<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Introduction to tridiagonal pairs<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: In this talk we consider a linear algebraic object called a tridiagonal pair. This object originated in algebraic graph theory, and has connections to orthogonal polynomials and representation theory. In our discussion we aim at a general mathematical audience; no prior experience with the above topics is assumed. Our main results are joint work with Tatsuro Ito and Kazumasa Nomura.<br />
<br />
The concept of a tridiagonal pair is best explained by starting with a special case called a Leonard pair. Let '''F''' denote a field, and let ''V'' denote a vector space over '''F''' with finite positive dimension. By a Leonard pair on ''V'' we mean a pair of linear transformations ''A''&nbsp;:&nbsp;''V''&nbsp;→&nbsp;''V'' and ''A''<sup>∗</sup>&nbsp;:&nbsp;''V''&nbsp;→&nbsp;''V'' that satisfy the following two conditions:<br />
#There exists a basis for ''V'' with respect to which the matrix representing ''A'' is diagonal and the matrix representing ''A''<sup>∗</sup> is irreducible tridiagonal;<br />
#There exists a basis for ''V'' with respect to which the matrix representing ''A''<sup>∗</sup> is diagonal and the matrix representing ''A'' is irreducible tridiagonal.<br />
<br />
We recall that a tridiagonal matrix is irreducible whenever each entry on the superdiagonal is nonzero and each entry on the subdiagonal is nonzero. The name "Leonard pair" is motivated by a connection to a 1982 theorem of the combinatorialist Doug Leonard involving the ''q''-Racah and related polynomials of the Askey scheme. Leonard’s theorem was heavily influenced by the work of Eiichi Bannai and Tatsuro Ito on ''P''- and ''Q''- polynomial association schemes, and the work of Richard Askey on orthogonal polynomials. These works in turn were influenced by the work of Philippe Delsarte on coding theory, dating from around 1973.<br />
<br />
The central result about Leonard pairs is that they are in bijection with the orthogonal polynomials that make up the terminating branch of the Askey scheme. This branch consists of the ''q''-Racah, ''q''-Hahn, dual ''q''-Hahn, ''q''-Krawtchouk, dual ''q''-Krawtchouk, quantum ''q''-Krawtchouk, affine ''q''-Krawtchouk, Racah, Hahn, dual-Hahn, Krawtchouk, and the Bannai/Ito polynomials. The bijection makes it possible to develop a uniform theory of these polynomials starting from the Leonard pair axiom, and we have done this over the past several years.<br />
<br />
A tridiagonal pair is a generalization of a Leonard pair and defined as follows. Let ''V'' denote a vector space over '''F''' with finite positive dimension. A ''tridiagonal pair on V'' is a pair of linear transformations ''A''&nbsp;:&nbsp;''V''&nbsp;→&nbsp;''V'' and ''A''<sup>∗</sup>&nbsp;:&nbsp;''V''&nbsp;→&nbsp;''V'' that satisfy the following four conditions:<br />
<ol><br />
<li>Each of ''A'', ''A''<sup>∗</sup> is diagonalizable on ''V'';<br />
<li>There exists an ordering {''V<sub>i</sub>''}<sub>''i''=0,...,''d''</sub> of the eigenspaces of ''A'' such that<br />
::''A''<sup>*</sup>''V<sub>i</sub>''&nbsp;⊆&nbsp;''V''<sub>''i''&minus;1</sub>&nbsp;+&nbsp;''V<sub>i</sub>''&nbsp;+&nbsp;''V''<sub>''i''+1</sub>&nbsp;&nbsp;(0&nbsp;≤&nbsp;''i''&nbsp;≤&nbsp;''d''),<br />
where ''V''<sub>&minus;1</sub>&nbsp;=&nbsp;0, ''V''<sub>''d''+1</sub>&nbsp;=&nbsp;0;<br />
<li>There exists an ordering {''V<sub>i</sub>''<sup>*</sup>}<sub>''i''=0,...,&delta;</sub> of the eigenspaces of ''A''<sup>*</sup> such that<br />
::''AV<sub>i</sub>''<sup>*</sup>&nbsp;⊆&nbsp;''V''<sup>*</sup><sub>''i''&minus;1</sub>&nbsp;+&nbsp;''V''<sup>*</sup><sub>''i''</sub>&nbsp;+&nbsp;''V''<sup>*</sup><sub>''i''+1</sub>&nbsp;&nbsp;(0&nbsp;≤&nbsp;''i''&nbsp;≤&nbsp;&delta;),<br />
where ''V''<sup>*</sup><sub>&minus;1</sub>&nbsp;=&nbsp;0, ''V''<sup>*</sup><sub>''d''+1</sub>&nbsp;=&nbsp;0;<br />
<li>There is no subspace ''W''&nbsp;⊆&nbsp;''V'' such that ''AW''&nbsp;⊆&nbsp;W, ''A''<sup>*</sup>''W''&nbsp;⊆&nbsp;''W'', ''W''&nbsp;&ne;&nbsp;0, ''W''&nbsp;&ne;&nbsp;''V''.<br />
</ol><br />
It turns out that ''d''&nbsp;=&nbsp;δ and this common value is called the diameter of the pair.<br />
A Leonard pair is the same thing as a tridiagonal pair for which the eigenspaces<br />
''V'' and ''V''<sup>∗</sup> all have dimension 1.<br />
<br />
Tridiagonal pairs arise naturally in the theory of ''P''- and ''Q''-polynomial association schemes, in connection with irreducible modules for the subconstituent algebra. They also appear in recent work of Pascal Baseilhac on the Ising model and related structures in statistical mechanics.<br />
<br />
In this talk we will summarize the basic facts about a tridiagonal pair, describing<br />
features such as the eigenvalues, dual eigenvalues, shape, tridiagonal relations,<br />
split decomposition, and parameter array. We will then focus on a special case<br />
said to be sharp and defined as follows. Referring to the tridiagonal pair ''A'', ''A''<sup>∗</sup> in the above definition, it turns out that for 0&nbsp;≤&nbsp;''i''&nbsp;≤&nbsp;''d'' the dimensions of ''V<sub>i</sub>'', ''V''<sub>''d''&minus;''i''</sub>, ''V''<sup>*</sup><sub>''i''</sub>, ''V''<sup>*</sup><sub>''d''&minus;''i''</sub> coincide; the pair ''A'', ''A''<sup>∗</sup> is called ''sharp'' whenever ''V''<sub>0</sub> has dimension 1. It is known that if '''F''' is algebraically closed then ''A'', ''A''<sup>∗</sup> is sharp.<br />
<br />
In our main result we classify the sharp tridiagonal pairs up to isomorphism.<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== March 29 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''David P. Roberts''' (U. Minnesota Morris)<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Lightly ramified number fields with Galois group ''S''.''M''<sub>12</sub>.''A''<br />
|-<br />
| bgcolor="#BCD2EE" |<br />
Abstract: Two of the most important invariants of an irreducible polynomial ''f''(''x'')&nbsp;&isin;&nbsp;'''Z'''[''x''&thinsp;] are its<br />
Galois group ''G'' and its field discriminant ''D''. The inverse Galois problem asks <br />
one to find a polynomial ''f''(''x'') having any prescribed Galois group ''G''. Refinements<br />
of this problem ask for ''D'' to be small in various senses, for example of the form<br />
&plusmn;&thinsp;''p<sup>a''</sup> for the smallest possible prime ''p''. <br />
<br />
The talk will discuss this problem in general, with a focus on the technique of <br />
specializing three-point covers for solving instances of it. Then it will pursue the cases of the <br />
Mathieu group ''M''<sub>12</sub>, its automorphism group ''M''<sub>12</sub>.2, its double cover<br />
2.''M''<sub>12</sub>, and the combined extension 2.''M''<sub>12</sub>.2. Among the polynomials<br />
found is<br />
{| style="background: #BCD2EE;" align="center"<br />
|-<br />
| ''f''(''x'')&nbsp;= || ''x''<sup>48</sup> + 2 ''e''<sup>3</sup> ''x''<sup>42</sup> + 69 ''e''<sup>5</sup> ''x''<sup>36</sup> + 868 ''e''<sup>7</sup> ''x''<sup>30</sup> &minus; 4174 ''e''<sup>7</sup> ''x''<sup>26</sup> + 11287 ''e''<sup>9</sup> ''x''<sup>24</sup><br />
|-<br />
| ||&minus; 4174 ''e''<sup>10</sup> ''x''<sup>20</sup> + 5340 ''e''<sup>12</sup> ''x''<sup>18</sup> + 131481 ''e''<sup>12</sup> ''x''<sup>14</sup> +17599 ''e''<sup>14</sup> ''x''<sup>12</sup> + 530098 ''e''<sup>14</sup> ''x''<sup>8</sup><br />
|-<br />
| ||+ 3910 ''e''<sup>16</sup> ''x''<sup>6</sup> + 4355569 ''e''<sup>14</sup> ''x''<sup>4</sup> + 20870 ''e''<sup>16</sup> ''x''<sup>2</sup> + 729 ''e''<sup>18</sup>,<br />
|}<br />
<br />
with ''e'' = 11. This polynomial has Galois group ''G''&nbsp;=&nbsp;2.''M''<sub>12</sub>.2 and<br />
field discriminant 11<sup>88</sup>. It makes ''M''<sub>12</sub> the<br />
first of the twenty-six sporadic simple groups &Gamma;<br />
known to have a polynomial with Galois group <br />
''G'' involving &Gamma; and field discriminant ''D'' <br />
the power of a single prime dividing |&Gamma;&thinsp;|.<br />
<br />
|} <br />
</center><br />
<br />
<br />
<br><br />
<br />
== April 12 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Chenyan Wu''' (Minnesota)<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Rallis inner product formula for theta lifts from metaplectic groups to orthogonal groups<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: Let π be a genuine cuspidal representation of the metaplectic group of rank<br />
''n''. We consider the theta lifts to the orthogonal group associated to a<br />
quadratic space of dimension 2''n''&nbsp;+&nbsp;1. We show a case of a regularized Rallis inner<br />
product formula that relates the pairing of theta lifts to the central value of the<br />
Langlands ''L''-function of π twisted by a genuine character. This enables us to<br />
demonstrate the relation between non-vanishing of theta lifts and the non-vanishing of<br />
central ''L''-values. We prove also a case of regularized Siegel–Weil formula which is<br />
missing in the literature, as it forms the basis of our proof of the Rallis inner<br />
product formula.<br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
<br />
==<span id="April 16"></span> April 16 (special day: '''Monday''', special time: '''3:30pm–4:30pm''', special place: '''VV B139''') ==<br />
<br />
<center><br />
<br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Hourong Qin''' (Nanjing U., China)<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: CM elliptic curves and quadratic polynomials representing primes <br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: No polynomial of degree two or higher has been proved to represent infinitely many primes. Let ''E'' be an elliptic curve defined over '''Q''' with complex multiplication. Fix an integer ''r''. We give sufficient and necessary conditions for ''a<sub>p''</sub>&nbsp;=&nbsp;''r'' for some prime ''p''. We show that there are infinitely many primes ''p'' such that ''a<sub>p''</sub>&nbsp;=&nbsp;''r'' for some fixed integer ''r'' if<br />
and only if a quadratic polynomial represents infinitely many primes ''p''.<br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== April 19 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Robert Guralnick''' (U. Southern California)<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: A variant of Burnside and Galois representations which are automorphic<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: Wiles, Taylor, Harris and others used the notion of a big<br />
representation of a finite<br />
group to show that certain representations are automorphic. Jack Thorne<br />
recently observed<br />
that one could weaken this notion of bigness to get the same conclusions. He<br />
called this property adequate. An absolutely irreducible representation ''V''<br />
of a finite group ''G'' in characteristic ''p'' is called adequate if ''G'' has<br />
no ''p''-quotients, the dimension<br />
of ''V'' is prime to ''p'', ''V'' has non-trivial self extensions and End(''V'') is<br />
generated by the linear<br />
span of the elements of order prime to ''p'' in ''G''. If ''G'' has order<br />
prime to ''p'', all of these conditions<br />
hold&mdash;the last condition is sometimes called Burnside's Lemma. We<br />
will discuss a recent<br />
result of Guralnick, Herzig, Taylor and Thorne showing that if ''p'' ><br />
2 dim ''V'' + 2, then<br />
any absolutely irreducible representation is adequate. We will also<br />
discuss some examples<br />
showing that the span of the ''p'''-elements in End(''V'') need not be all of End(''V'').<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== April 26 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Frank Thorne''' (U. South Carolina)<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Secondary terms in counting functions for cubic fields<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
<br />
Abstract: We will discuss our proof of secondary terms of order ''X''<sup>5/6</sup> in the Davenport–Heilbronn<br />
theorems on cubic fields and 3-torsion in class groups of quadratic fields. For cubic<br />
fields this confirms a conjecture of Datskovsky–Wright and Roberts. We also will describe<br />
some generalizations, in particular to arithmetic progressions, where we discover a<br />
curious bias in the secondary term.<br />
<br />
Roberts’ conjecture has also been proved independently by Bhargava, Shankar, and<br />
Tsimerman. Their proof uses the geometry of numbers, while our proof uses the analytic<br />
theory of Shintani zeta functions.<br />
<br />
We will also discuss a combined approach which yields further improved error terms. If<br />
there is time (or after the talk), I will also discuss a couple of side projects and my<br />
plans for further related work.<br />
<br />
This is joint work with Takashi Taniguchi.<br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== May 3 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Alina Cojocaru''' (U. Illinois at Chicago)<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Frobenius fields for elliptic curves<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: Let ''E'' be an elliptic curve defined over '''Q'''. For a prime p of good reduction for ''E'', let &pi;<sub>p</sub> be the p-Weil root of E and Q(pi<sub>p</sub>) the associated imaginary quadratic field generated by pi_<sub>p</sub>. In 1976, Serge Lang and Hale Trotter formulated a conjectural asymptotic formula for the number of primes ''p < x'' for which '''Q'''(pi_<sub>p</sub>) is isomorphic to a fixed imaginary quadratic field. I will discuss progress on this conjecture, in particular an average result confirming the predicted asymptotic formula. This is joint work with Henryk Iwaniec and Nathan Jones. <br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== May 10 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Samit Dasgupta''' (UC Santa Cruz)<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: tba<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: tba <br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== Organizer contact information ==<br />
<br />
[http://www.math.wisc.edu/~shamgar/ Shamgar Gurevich]<br />
<br />
[http://www.math.wisc.edu/~rharron/ Robert Harron]<br />
<br />
[http://www.math.wisc.edu/~klagsbru/ Zev Klagsbrun]<br />
<br />
[http://www.math.wisc.edu/~mmwood/ Melanie Matchett Wood]<br />
<br><br />
<br />
<br />
<br />
----<br />
Return to the [[NTS|Number Theory Seminar Page]]<br />
<br />
Return to the [[Algebra|Algebra Group Page]]</div>Klagsbruhttps://wiki.math.wisc.edu/index.php?title=NTS_Spring_2012&diff=3762NTS Spring 20122012-04-19T16:37:00Z<p>Klagsbru: /* Spring 2012 Semester */</p>
<hr />
<div>= Number Theory – Representation Theory Seminar, University of Wisconsin–Madison =<br />
<br />
<br />
*'''When:''' Thursdays, 2:30pm–3:30pm.<br />
*'''Where:''' Van Vleck Hall B129 <br />
*Please join the [https://mailhost.math.wisc.edu/mailman/listinfo/nts NT/RT mailing list:] (you must be on a math department computer to use this link).<br />
<br />
= Spring 2012 Semester =<br />
<br />
<center><br />
<br />
{| style="color:black; font-size:120%" border="0" cellpadding="14" cellspacing="5"<br />
|-<br />
| bgcolor="#D0D0D0" width="300" align="center"|'''Date'''<br />
| bgcolor="#F0A0A0" width="300" align="center"|'''Speaker'''<br />
| bgcolor="#BCD2EE" width="300" align="center"|'''Title (click to see abstract)'''<br />
|-<br />
| bgcolor="#E0E0E0"| Feb 2 (Thurs.)<br />
| bgcolor="#F0B0B0"| [http://www.math.wisc.edu/~dummit/ Evan Dummit] <br> (Madison)<br />
| bgcolor="#BCE2FE"|[[NTS Spring 2012/Abstracts#February 2 | <font color="black"><em>Kakeya sets over non-archimedean local rings</em></font>]]<br />
|- <br />
| bgcolor="#E0E0E0"| Feb 16 (Thurs.)<br />
| bgcolor="#F0B0B0"|[http://www.math.wisc.edu/~thyang/ Tonghai Yang] <br> (Madison)<br />
| bgcolor="#BCE2FE"|[[NTS Spring 2012/Abstracts#February 16 | <font color="black"><em>A little linear algebra on CM abelian surfaces </em></font>]]<br />
|- <br />
| bgcolor="#E0E0E0"| Feb 23 (Thurs.)<br />
| bgcolor="#F0B0B0"| [http://www.math.wisc.edu/~vincent/ Christelle Vincent] <br> (Madison)<br />
| bgcolor="#BCE2FE"|[[NTS Spring 2012/Abstracts#February 23 | <font color="black"><em>Drinfeld modular forms</em></font>]]<br />
|- <br />
| bgcolor="#E0E0E0"| Mar 1 (Thurs.)<br />
| bgcolor="#F0B0B0"| [http://www.math.wisc.edu/~shamgar/ Shamgar Gurevich] <br> (Madison)<br />
| bgcolor="#BCE2FE"|[[NTS Spring 2012/Abstracts#March 1 | <font color="black"><em>Computing the matched filter in linear time</em></font>]]<br />
|- <br />
| bgcolor="#E0E0E0"| Mar 8 (Thurs.)<br />
| bgcolor="#F0B0B0"| [http://www.math.wisc.edu/~klagsbru/ Zev Klagsbrun] <br> (Madison)<br />
| bgcolor="#BCE2FE"|[[NTS Spring 2012/Abstracts#March 8 | <font color="black"><em>Erdős–Kac type theorems</em></font>]]<br />
|- <br />
| bgcolor="#E0E0E0"| Mar 15 (Thurs.)<br />
| bgcolor="#F0B0B0"| [http://www.math.wisc.edu/~yqzhao/ Yongqiang Zhao] <br> (Madison)<br />
| bgcolor="#BCE2FE"|[[NTS Spring 2012/Abstracts#March 15 | <font color="black"><em>On the Roberts conjecture</em></font>]]<br />
|- <br />
| bgcolor="#E0E0E0"| Mar 22 (Thurs.)<br />
| bgcolor="#F0B0B0"| [http://www.math.wisc.edu/~terwilli/ Paul Terwilliger] <br> (Madison)<br />
| bgcolor="#BCE2FE"|[[NTS Spring 2012/Abstracts#March 22 | <font color="black"><em>Introduction to tridiagonal pairs</em></font>]]<br />
|- <br />
| bgcolor="#E0E0E0"| Mar 29 (Thurs.)<br />
| bgcolor="#F0B0B0"| [http://cda.morris.umn.edu/~roberts/ David P. Roberts]<br> (U. of Minnesota Morris)<br />
| bgcolor="#BCE2FE"|[[NTS Spring 2012/Abstracts#March 29 | <font color="black"><em>Lightly ramified number fields with Galois group </em>''S''.''M''<sub>12</sub>.''A''</font>]]<br />
|- <br />
| bgcolor="#E0E0E0"| April 5 (Thurs.)<br />
| bgcolor="#F0B0B0"| No seminar <br> (Spring break!)<br />
| bgcolor="#BCE2FE"|[[NTS Spring 2012/Abstracts#April 5 | <font color="black"><em>Spring break!</em></font>]]<br />
|- <br />
| bgcolor="#E0E0E0"| April 12 (Thurs.)<br />
| bgcolor="#F0B0B0"| [http://math.umn.edu/~cywu/ Chenyan Wu]<br> (Minnesota)<br />
| bgcolor="#BCE2FE"|[[NTS Spring 2012/Abstracts#April 12 | <font color="black"><em>Rallis inner product formula for theta lifts from metaplectic groups to orthogonal groups</em></font>]]<br />
|- <br />
| bgcolor="#E0E0E0"| April 16 – '''Monday Special'''<br><br>'''3:30pm–4:30pm''' in Van Vleck '''B139'''<br />
| bgcolor="#F0B0B0"| Hourong Qin<br> (Nanjing University, China)<br />
| bgcolor="#BCE2FE"|[[NTS Spring 2012/Abstracts#April 16 | <font color="black"><em>CM elliptic curves and quadratic polynomials representing primes </em></font>]]<br />
|-<br />
| bgcolor="#E0E0E0"| April 19 (Thurs.)<br />
| bgcolor="#F0B0B0"| [http://www-bcf.usc.edu/~guralnic/ Robert Guralnick] <br>(U. of Southern California) <br />
| bgcolor="#BCE2FE"|[[NTS Spring 2012/Abstracts#April 19 | <font color="black"><em>A variant of Burnside and Galois representations which are automorphic </em></font>]]<br />
|- <br />
| bgcolor="#E0E0E0"| April 26 (Thurs.)<br />
| bgcolor="#F0B0B0"|[http://www.math.sc.edu/~thornef/ Frank Thorne] <br> (U. South Carolina)<br />
| bgcolor="#BCE2FE"|[[NTS Spring 2012/Abstracts#April 26 | <font color="black"><em>Secondary terms in counting functions for cubic fields</em></font>]]<br />
|- <br />
| bgcolor="#E0E0E0"| May 3 (Thurs.)<br />
| bgcolor="#F0B0B0"| [http://homepages.math.uic.edu/~cojocaru/ Alina Cojocaru] <br> (U. Illinois at Chicago)<br />
| bgcolor="#BCE2FE"|[[NTS Spring 2012/Abstracts#May 3 | <font color="black"><em> Frobenius fields for elliptic curves</em></font>]]<br />
|- <br />
| bgcolor="#E0E0E0"| May 10 (Thurs.)<br />
| bgcolor="#F0B0B0"| [http://people.ucsc.edu/~sdasgup2/ Samit Dasgupta] <br> (UC Santa Cruz)<br />
| bgcolor="#BCE2FE"|[[NTS Spring 2012/Abstracts | <font color="black"><em>TBA</em></font>]]<br />
|}<br />
<br />
</center><br />
<br />
<br><br />
<br />
= Organizer contact information =<br />
[http://www.math.wisc.edu/~shamgar/ Shamgar Gurevich]<br />
<br />
[http://www.math.wisc.edu/~rharron/ Robert Harron]<br />
<br />
[http://www.math.wisc.edu/~klagsbru/ Zev Klagsbrun]<br />
<br />
[http://www.math.wisc.edu/~mmwood/ Melanie Matchett Wood]<br />
<br />
<br />
<br />
----<br />
Also of interest is the [[NTSGrad|Grad student seminar]] which meets on Tuesdays.<br><br />
Last semester's seminar page is [[NTS Fall 2011|here]].<br />
----<br />
Return to the [[Algebra|Algebra Group Page]]</div>Klagsbruhttps://wiki.math.wisc.edu/index.php?title=NTS_Spring_2012/Abstracts&diff=3761NTS Spring 2012/Abstracts2012-04-19T16:36:45Z<p>Klagsbru: /* May 3 */</p>
<hr />
<div>== February 2 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Evan Dummit''' (Madison)<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Kakeya sets over non-archimedean local rings<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: In a forthcoming paper with Marci Habliscek, we constructed a Kakeya set over the formal power series ring '''F'''<sub>''q''</sub><nowiki>[[</nowiki>''t''&thinsp;<nowiki>]]</nowiki>, answering a question posed by Ellenberg, Oberlin, and Tao. My talk will be devoted to explaining some of the older history of the Kakeya problem in analysis and the newer history of the Kakeya problem in combinatorics, including my joint work with Marci. In particular, I will give Dvir's solution of the Kakeya problem over finite fields, and explain the problem's extension to other classes of rings. <br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== February 16 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Tonghai Yang''' (Madison)<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: A little linear algebra on CM abelian surfaces<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: In this talk, I will discuss an interesting Hermitian form structure on the space of ''special endormorphisms'' of a CM abelian surface, and how to use it make a moduli problem and prove an arithmetic Siegel–Weil formula over a real quadratic field. This is a joint work with Ben Howard.<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== February 23 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Christelle Vincent''' (Madison)<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Drinfeld modular forms<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: We will begin by introducing the Drinfeld setting, and in particular Drinfeld modular forms and their connection to the geometry of Drinfeld modular curves. We will then present some results about Drinfeld modular forms that we obtained in the process of computing certain geometric points on Drinfeld modular curves. More precisely, we will talk about Drinfeld modular forms modulo ''P'', for ''P'' a prime ideal in '''F'''<sub>''q''</sub>[''T''&thinsp;], and about Drinfeld quasi-modular forms.<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== March 1 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Shamgar Gurevich''' (Madison)<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Computing the Matched Filter in Linear Time<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: In the digital radar problem we design a function (waveform) S(t) in the Hilbert space H&nbsp;=&nbsp;C('''Z'''/''p'') of complex valued functions on '''Z'''/''p''&nbsp;=&nbsp;{0,&nbsp;...,&nbsp;''p''&nbsp;&minus;&nbsp;1}, the integers modulo a prime number ''p''&nbsp;≫&nbsp;0. We transmit the function S(t) using the radar to the object that we want to detect. The wave S(t) hits the object, and is reflected back via the echo wave R(t) in H, which has the form <br />
<br />
R(t)&nbsp;=&nbsp;exp{2πiωt/''p''}⋅S(t+τ)&nbsp;+&nbsp;W(t), <br />
<br />
where W(t) in H is a white noise, and τ,&nbsp;ω in '''Z'''/''p'', encode the distance from, and velocity of, the object. <br />
<br />
Problem (digital radar problem) Extract τ,&nbsp;ω from R and S. <br />
<br />
In the lecture I first introduce the classical matched filter (MF) algorithm that suggests the 'traditional' way (using fast Fourier transform) to solve the digital radar problem in order of ''p''<sup>2</sup>⋅log(''p'') operations. I will then explain how to use techniques from group representation theory and arithmetic to design (construct) waveforms S(t) which enable us to introduce a fast matched filter (FMF) algorithm, that we call the "flag algorithm", which solves the digital radar problem in a much faster way of order of ''p''⋅log(''p'') operations. <br />
<br />
I will demonstrate additional applications to mobile communication, and global positioning system (GPS). <br />
<br />
This is a joint work with A. Fish (Math, Madison), R. Hadani (Math, Austin), A. Sayeed (Electrical Engineering, Madison), and O. Schwartz (Electrical Engineering and Computer Science, Berkeley). <br />
<br />
The lecture is suitable for general math/engineering audience.<br />
<br />
|} <br />
</center><br />
<br />
<br><br />
== March 8 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Zev Klagsbrun''' (Madison)<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Erdős–Kac Type Theorems<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: In its most popular formulation, the Erdős–Kac Theorem gives a distribution on the number of distinct primes factors (ω(''n'')) of the numbers up to ''N''. Variants of the Erdős–Kac Theorem yield distributions on additive functions in a surprising number of settings. This talk will outline the basics of the theory by focusing on some results of Granville and Soundararajan that allow one to easily prove Erdős–Kac type results for a variety of problems as well as present a recent result of my own using the Granville and Soundararajan framework.<br />
<br />
The lecture is suitable for general math audience.<br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== March 15 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Yongqiang Zhao''' (Madison)<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: On the Roberts conjecture<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: Let ''N''(''X'')&nbsp;=&nbsp;#{''K''&nbsp;<nowiki>|</nowiki>&nbsp;[''K'':'''Q''']&nbsp;=&nbsp;3,&nbsp;disc(''K'')&nbsp;≤&nbsp;''X''} be the counting function of cubic fields of bounded<br />
discriminant. The Roberts Conjecture is about the second term of this counting function. This conjecture was thought to be hard and dormant for some time. However, recently, four very different<br />
approaches to this problem were developed independently by Bhargava, Shankar and Tsimerman,<br />
Hough, Taniguchi and Thorne, and myself.<br />
In this talk, I will mention the first three approaches very briefly, then focus on my results on the function field case. I will give an outline of the proof. If time permits, I will indicate how the geometry<br />
feeds back to the number field case, in particular, how one could possibly define a new invariant<br />
for cubic fields.<br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== March 22 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Paul Terwilliger''' (Madison)<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Introduction to tridiagonal pairs<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: In this talk we consider a linear algebraic object called a tridiagonal pair. This object originated in algebraic graph theory, and has connections to orthogonal polynomials and representation theory. In our discussion we aim at a general mathematical audience; no prior experience with the above topics is assumed. Our main results are joint work with Tatsuro Ito and Kazumasa Nomura.<br />
<br />
The concept of a tridiagonal pair is best explained by starting with a special case called a Leonard pair. Let '''F''' denote a field, and let ''V'' denote a vector space over '''F''' with finite positive dimension. By a Leonard pair on ''V'' we mean a pair of linear transformations ''A''&nbsp;:&nbsp;''V''&nbsp;→&nbsp;''V'' and ''A''<sup>∗</sup>&nbsp;:&nbsp;''V''&nbsp;→&nbsp;''V'' that satisfy the following two conditions:<br />
#There exists a basis for ''V'' with respect to which the matrix representing ''A'' is diagonal and the matrix representing ''A''<sup>∗</sup> is irreducible tridiagonal;<br />
#There exists a basis for ''V'' with respect to which the matrix representing ''A''<sup>∗</sup> is diagonal and the matrix representing ''A'' is irreducible tridiagonal.<br />
<br />
We recall that a tridiagonal matrix is irreducible whenever each entry on the superdiagonal is nonzero and each entry on the subdiagonal is nonzero. The name "Leonard pair" is motivated by a connection to a 1982 theorem of the combinatorialist Doug Leonard involving the ''q''-Racah and related polynomials of the Askey scheme. Leonard’s theorem was heavily influenced by the work of Eiichi Bannai and Tatsuro Ito on ''P''- and ''Q''- polynomial association schemes, and the work of Richard Askey on orthogonal polynomials. These works in turn were influenced by the work of Philippe Delsarte on coding theory, dating from around 1973.<br />
<br />
The central result about Leonard pairs is that they are in bijection with the orthogonal polynomials that make up the terminating branch of the Askey scheme. This branch consists of the ''q''-Racah, ''q''-Hahn, dual ''q''-Hahn, ''q''-Krawtchouk, dual ''q''-Krawtchouk, quantum ''q''-Krawtchouk, affine ''q''-Krawtchouk, Racah, Hahn, dual-Hahn, Krawtchouk, and the Bannai/Ito polynomials. The bijection makes it possible to develop a uniform theory of these polynomials starting from the Leonard pair axiom, and we have done this over the past several years.<br />
<br />
A tridiagonal pair is a generalization of a Leonard pair and defined as follows. Let ''V'' denote a vector space over '''F''' with finite positive dimension. A ''tridiagonal pair on V'' is a pair of linear transformations ''A''&nbsp;:&nbsp;''V''&nbsp;→&nbsp;''V'' and ''A''<sup>∗</sup>&nbsp;:&nbsp;''V''&nbsp;→&nbsp;''V'' that satisfy the following four conditions:<br />
<ol><br />
<li>Each of ''A'', ''A''<sup>∗</sup> is diagonalizable on ''V'';<br />
<li>There exists an ordering {''V<sub>i</sub>''}<sub>''i''=0,...,''d''</sub> of the eigenspaces of ''A'' such that<br />
::''A''<sup>*</sup>''V<sub>i</sub>''&nbsp;⊆&nbsp;''V''<sub>''i''&minus;1</sub>&nbsp;+&nbsp;''V<sub>i</sub>''&nbsp;+&nbsp;''V''<sub>''i''+1</sub>&nbsp;&nbsp;(0&nbsp;≤&nbsp;''i''&nbsp;≤&nbsp;''d''),<br />
where ''V''<sub>&minus;1</sub>&nbsp;=&nbsp;0, ''V''<sub>''d''+1</sub>&nbsp;=&nbsp;0;<br />
<li>There exists an ordering {''V<sub>i</sub>''<sup>*</sup>}<sub>''i''=0,...,&delta;</sub> of the eigenspaces of ''A''<sup>*</sup> such that<br />
::''AV<sub>i</sub>''<sup>*</sup>&nbsp;⊆&nbsp;''V''<sup>*</sup><sub>''i''&minus;1</sub>&nbsp;+&nbsp;''V''<sup>*</sup><sub>''i''</sub>&nbsp;+&nbsp;''V''<sup>*</sup><sub>''i''+1</sub>&nbsp;&nbsp;(0&nbsp;≤&nbsp;''i''&nbsp;≤&nbsp;&delta;),<br />
where ''V''<sup>*</sup><sub>&minus;1</sub>&nbsp;=&nbsp;0, ''V''<sup>*</sup><sub>''d''+1</sub>&nbsp;=&nbsp;0;<br />
<li>There is no subspace ''W''&nbsp;⊆&nbsp;''V'' such that ''AW''&nbsp;⊆&nbsp;W, ''A''<sup>*</sup>''W''&nbsp;⊆&nbsp;''W'', ''W''&nbsp;&ne;&nbsp;0, ''W''&nbsp;&ne;&nbsp;''V''.<br />
</ol><br />
It turns out that ''d''&nbsp;=&nbsp;δ and this common value is called the diameter of the pair.<br />
A Leonard pair is the same thing as a tridiagonal pair for which the eigenspaces<br />
''V'' and ''V''<sup>∗</sup> all have dimension 1.<br />
<br />
Tridiagonal pairs arise naturally in the theory of ''P''- and ''Q''-polynomial association schemes, in connection with irreducible modules for the subconstituent algebra. They also appear in recent work of Pascal Baseilhac on the Ising model and related structures in statistical mechanics.<br />
<br />
In this talk we will summarize the basic facts about a tridiagonal pair, describing<br />
features such as the eigenvalues, dual eigenvalues, shape, tridiagonal relations,<br />
split decomposition, and parameter array. We will then focus on a special case<br />
said to be sharp and defined as follows. Referring to the tridiagonal pair ''A'', ''A''<sup>∗</sup> in the above definition, it turns out that for 0&nbsp;≤&nbsp;''i''&nbsp;≤&nbsp;''d'' the dimensions of ''V<sub>i</sub>'', ''V''<sub>''d''&minus;''i''</sub>, ''V''<sup>*</sup><sub>''i''</sub>, ''V''<sup>*</sup><sub>''d''&minus;''i''</sub> coincide; the pair ''A'', ''A''<sup>∗</sup> is called ''sharp'' whenever ''V''<sub>0</sub> has dimension 1. It is known that if '''F''' is algebraically closed then ''A'', ''A''<sup>∗</sup> is sharp.<br />
<br />
In our main result we classify the sharp tridiagonal pairs up to isomorphism.<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== March 29 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''David P. Roberts''' (U. Minnesota Morris)<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Lightly ramified number fields with Galois group ''S''.''M''<sub>12</sub>.''A''<br />
|-<br />
| bgcolor="#BCD2EE" |<br />
Abstract: Two of the most important invariants of an irreducible polynomial ''f''(''x'')&nbsp;&isin;&nbsp;'''Z'''[''x''&thinsp;] are its<br />
Galois group ''G'' and its field discriminant ''D''. The inverse Galois problem asks <br />
one to find a polynomial ''f''(''x'') having any prescribed Galois group ''G''. Refinements<br />
of this problem ask for ''D'' to be small in various senses, for example of the form<br />
&plusmn;&thinsp;''p<sup>a''</sup> for the smallest possible prime ''p''. <br />
<br />
The talk will discuss this problem in general, with a focus on the technique of <br />
specializing three-point covers for solving instances of it. Then it will pursue the cases of the <br />
Mathieu group ''M''<sub>12</sub>, its automorphism group ''M''<sub>12</sub>.2, its double cover<br />
2.''M''<sub>12</sub>, and the combined extension 2.''M''<sub>12</sub>.2. Among the polynomials<br />
found is<br />
{| style="background: #BCD2EE;" align="center"<br />
|-<br />
| ''f''(''x'')&nbsp;= || ''x''<sup>48</sup> + 2 ''e''<sup>3</sup> ''x''<sup>42</sup> + 69 ''e''<sup>5</sup> ''x''<sup>36</sup> + 868 ''e''<sup>7</sup> ''x''<sup>30</sup> &minus; 4174 ''e''<sup>7</sup> ''x''<sup>26</sup> + 11287 ''e''<sup>9</sup> ''x''<sup>24</sup><br />
|-<br />
| ||&minus; 4174 ''e''<sup>10</sup> ''x''<sup>20</sup> + 5340 ''e''<sup>12</sup> ''x''<sup>18</sup> + 131481 ''e''<sup>12</sup> ''x''<sup>14</sup> +17599 ''e''<sup>14</sup> ''x''<sup>12</sup> + 530098 ''e''<sup>14</sup> ''x''<sup>8</sup><br />
|-<br />
| ||+ 3910 ''e''<sup>16</sup> ''x''<sup>6</sup> + 4355569 ''e''<sup>14</sup> ''x''<sup>4</sup> + 20870 ''e''<sup>16</sup> ''x''<sup>2</sup> + 729 ''e''<sup>18</sup>,<br />
|}<br />
<br />
with ''e'' = 11. This polynomial has Galois group ''G''&nbsp;=&nbsp;2.''M''<sub>12</sub>.2 and<br />
field discriminant 11<sup>88</sup>. It makes ''M''<sub>12</sub> the<br />
first of the twenty-six sporadic simple groups &Gamma;<br />
known to have a polynomial with Galois group <br />
''G'' involving &Gamma; and field discriminant ''D'' <br />
the power of a single prime dividing |&Gamma;&thinsp;|.<br />
<br />
|} <br />
</center><br />
<br />
<br />
<br><br />
<br />
== April 12 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Chenyan Wu''' (Minnesota)<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Rallis inner product formula for theta lifts from metaplectic groups to orthogonal groups<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: Let π be a genuine cuspidal representation of the metaplectic group of rank<br />
''n''. We consider the theta lifts to the orthogonal group associated to a<br />
quadratic space of dimension 2''n''&nbsp;+&nbsp;1. We show a case of a regularized Rallis inner<br />
product formula that relates the pairing of theta lifts to the central value of the<br />
Langlands ''L''-function of π twisted by a genuine character. This enables us to<br />
demonstrate the relation between non-vanishing of theta lifts and the non-vanishing of<br />
central ''L''-values. We prove also a case of regularized Siegel–Weil formula which is<br />
missing in the literature, as it forms the basis of our proof of the Rallis inner<br />
product formula.<br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
<br />
==<span id="April 16"></span> April 16 (special day: '''Monday''', special time: '''3:30pm–4:30pm''', special place: '''VV B139''') ==<br />
<br />
<center><br />
<br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Hourong Qin''' (Nanjing U., China)<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: CM elliptic curves and quadratic polynomials representing primes <br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: No polynomial of degree two or higher has been proved to represent infinitely many primes. Let ''E'' be an elliptic curve defined over '''Q''' with complex multiplication. Fix an integer ''r''. We give sufficient and necessary conditions for ''a<sub>p''</sub>&nbsp;=&nbsp;''r'' for some prime ''p''. We show that there are infinitely many primes ''p'' such that ''a<sub>p''</sub>&nbsp;=&nbsp;''r'' for some fixed integer ''r'' if<br />
and only if a quadratic polynomial represents infinitely many primes ''p''.<br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== April 19 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Robert Guralnick''' (U. Southern California)<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: A variant of Burnside and Galois representations which are automorphic<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: Wiles, Taylor, Harris and others used the notion of a big<br />
representation of a finite<br />
group to show that certain representations are automorphic. Jack Thorne<br />
recently observed<br />
that one could weaken this notion of bigness to get the same conclusions. He<br />
called this property adequate. An absolutely irreducible representation ''V''<br />
of a finite group ''G'' in characteristic ''p'' is called adequate if ''G'' has<br />
no ''p''-quotients, the dimension<br />
of ''V'' is prime to ''p'', ''V'' has non-trivial self extensions and End(''V'') is<br />
generated by the linear<br />
span of the elements of order prime to ''p'' in ''G''. If ''G'' has order<br />
prime to ''p'', all of these conditions<br />
hold&mdash;the last condition is sometimes called Burnside's Lemma. We<br />
will discuss a recent<br />
result of Guralnick, Herzig, Taylor and Thorne showing that if ''p'' ><br />
2 dim ''V'' + 2, then<br />
any absolutely irreducible representation is adequate. We will also<br />
discuss some examples<br />
showing that the span of the ''p'''-elements in End(''V'') need not be all of End(''V'').<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== April 26 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Frank Thorne''' (U. South Carolina)<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Secondary terms in counting functions for cubic fields<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
<br />
Abstract: We will discuss our proof of secondary terms of order ''X''<sup>5/6</sup> in the Davenport–Heilbronn<br />
theorems on cubic fields and 3-torsion in class groups of quadratic fields. For cubic<br />
fields this confirms a conjecture of Datskovsky–Wright and Roberts. We also will describe<br />
some generalizations, in particular to arithmetic progressions, where we discover a<br />
curious bias in the secondary term.<br />
<br />
Roberts’ conjecture has also been proved independently by Bhargava, Shankar, and<br />
Tsimerman. Their proof uses the geometry of numbers, while our proof uses the analytic<br />
theory of Shintani zeta functions.<br />
<br />
We will also discuss a combined approach which yields further improved error terms. If<br />
there is time (or after the talk), I will also discuss a couple of side projects and my<br />
plans for further related work.<br />
<br />
This is joint work with Takashi Taniguchi.<br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== May 3 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Alina Cojocaru''' (U. Illinois at Chicago)<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Frobenius fields for elliptic curves<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: Let E be an elliptic curve defined over Q. For a prime p of good reduction for E, let pi_p be the p-Weil root of E and Q(pi_p) the associated imaginary quadratic field generated by pi_p. In 1976, Serge Lang and Hale Trotter formulated a conjectural asymptotic formula for the number of primes p < x for which Q(pi_p) is isomorphic to a fixed imaginary quadratic field. I will discuss progress on this conjecture, in particular an average result confirming the predicted asymptotic formula. This is joint work with Henryk Iwaniec and Nathan Jones. <br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== May 10 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Samit Dasgupta''' (UC Santa Cruz)<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: tba<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: tba <br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== Organizer contact information ==<br />
<br />
[http://www.math.wisc.edu/~shamgar/ Shamgar Gurevich]<br />
<br />
[http://www.math.wisc.edu/~rharron/ Robert Harron]<br />
<br />
[http://www.math.wisc.edu/~klagsbru/ Zev Klagsbrun]<br />
<br />
[http://www.math.wisc.edu/~mmwood/ Melanie Matchett Wood]<br />
<br><br />
<br />
<br />
<br />
----<br />
Return to the [[NTS|Number Theory Seminar Page]]<br />
<br />
Return to the [[Algebra|Algebra Group Page]]</div>Klagsbruhttps://wiki.math.wisc.edu/index.php?title=NTS_Spring_2012/Abstracts&diff=3581NTS Spring 2012/Abstracts2012-02-27T20:31:21Z<p>Klagsbru: </p>
<hr />
<div>== February 2 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Evan Dummit''' (Madison)<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Kakeya sets over non-archimedean local rings<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: In a forthcoming paper with Marci Habliscek, we constructed a Kakeya set over the formal power series ring '''F'''<sub>''q''</sub><nowiki>[[</nowiki>''t''&thinsp;<nowiki>]]</nowiki>, answering a question posed by Ellenberg, Oberlin, and Tao. My talk will be devoted to explaining some of the older history of the Kakeya problem in analysis and the newer history of the Kakeya problem in combinatorics, including my joint work with Marci. In particular, I will give Dvir's solution of the Kakeya problem over finite fields, and explain the problem's extension to other classes of rings. <br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== February 16 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Tonghai Yang''' (Madison)<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: A little linear algebra on CM abelian surfaces<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: In this talk, I will discuss an interesting Hermitian form structure on the space of ''special endormorphisms'' of a CM abelian surface, and how to use it make a moduli problem and prove an arithmetic Siegel–Weil formula over a real quadratic field. This is a joint work with Ben Howard.<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== February 23 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Christelle Vincent''' (Madison)<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Drinfeld modular forms<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: We will begin by introducing the Drinfeld setting, and in particular Drinfeld modular forms and their connection to the geometry of Drinfeld modular curves. We will then present some results about Drinfeld modular forms that we obtained in the process of computing certain geometric points on Drinfeld modular curves. More precisely, we will talk about Drinfeld modular forms modulo ''P'', for ''P'' a prime ideal in '''F'''<sub>''q''</sub>[''T''&thinsp;], and about Drinfeld quasi-modular forms.<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== March 1 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Shamgar Gurevich''' (Madison)<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Computing the Matched Filter in Linear Time<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: In the digital radar problem we design a function (waveform) S(t) in the Hilbert space H&nbsp;=&nbsp;C('''Z'''/''p'') of complex valued functions on '''Z'''/''p''&nbsp;=&nbsp;{0,&nbsp;...,&nbsp;''p''&nbsp;&minus;&nbsp;1}, the integers modulo a prime number ''p''&nbsp;≫&nbsp;0. We transmit the function S(t) using the radar to the object that we want to detect. The wave S(t) hits the object, and is reflected back via the echo wave R(t) in H, which has the form <br />
<br />
R(t)&nbsp;=&nbsp;exp{2πiωt/''p''}⋅S(t+τ)&nbsp;+&nbsp;W(t), <br />
<br />
where W(t) in H is a white noise, and τ,&nbsp;ω in '''Z'''/''p'', encode the distance from, and velocity of, the object. <br />
<br />
Problem (digital radar problem) Extract τ,&nbsp;ω from R and S. <br />
<br />
In the lecture I first introduce the classical matched filter (MF) algorithm that suggests the 'traditional' way (using fast Fourier transform) to solve the digital radar problem in order of ''p''<sup>2</sup>⋅log(''p'') operations. I will then explain how to use techniques from group representation theory and arithmetic to design (construct) waveforms S(t) which enable us to introduce a fast matched filter (FMF) algorithm, that we call the "flag algorithm", which solves the digital radar problem in a much faster way of order of ''p''⋅log(''p'') operations. <br />
<br />
I will demonstrate additional applications to mobile communication, and global positioning system (GPS). <br />
<br />
This is a joint work with A. Fish (Math, Madison), R. Hadani (Math, Austin), A. Sayeed (Electrical Engineering, Madison), and O. Schwartz (Electrical Engineering and Computer Science, Berkeley). <br />
<br />
The lecture is suitable for general math/engineering audience.<br />
<br />
|} <br />
</center><br />
<br />
<br><br />
== March 8 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Zev Klagsbrun''' (Madison)<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Erdos-Kac Type Theorems<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: In it's most popular formulation, the Erdos-Kac Theorem gives a distribution on the number of distinct primes factors (''ω(n)'') of the numbers up to ''N''. Variants of the Erdos-Kac Theorem yield distributions on additive functions in a surprising number of settings. This talk will outline the basics of the theory by focusing on some results of Granville and Soundararajan that allow one to easily prove Erdos-Kac type results for a variety of problems as well as present a recent result of my own using the Granville and Soundararajan framework.<br />
<br />
The lecture is suitable for general math audience.<br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== March 15 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Yongqiang Zhao''' (Madison)<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: tba<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: tba<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== March 22 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Paul Terwilliger''' (Madison)<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: tba<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: tba<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
<!--== September 29 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Nigel Boston''' (Madison)<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Non-abelian Cohen-Lenstra heuristics.<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: In 1983, Cohen and Lenstra observed that the frequency with which a given abelian ''p''-group ''A'' (''p'' odd) arises as the ''p''-class group of an imaginary quadratic field ''K'' is apparently proportional to 1/|Aut(''A'')|. The Galois group of the maximal<br />
unramified ''p''-extension of ''K'' has abelianization ''A'' and one might then ask how frequently a given ''p''-group ''G'' arises. We<br />
develop a theory wherein this frequency is inversely proportional to the size of its automorphism group in a new category<br />
and then test this against computations. If time permits, I shall describe progress on the real quadratic case. This is<br />
joint work with Michael Bush and Farshid Hajir.<br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== October 6 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Zhiwei Yun''' (MIT)<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Exceptional Lie groups as motivic Galois groups<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: More than two decades ago, Serre asked the following<br />
question: can exceptional Lie groups be realized as the motivic Galois<br />
group of some motive over a number field? The question has been open<br />
for exceptional groups other than ''G''<sub>2</sub>. In this talk, I will show how<br />
to use geometric Langlands theory to give a uniform construction of<br />
motives with motivic Galois groups ''E''<sub>7</sub>, ''E''<sub>8</sub> and ''G''<sub>2</sub>, hence giving an<br />
affirmative answer to Serre's question in these cases.<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== October 13 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Melanie Matchett Wood''' (Madison)<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: The probability that a curve over a finite field is smooth<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
<br />
Abstract: Given a fixed variety over a finite field, we ask what<br />
proportion of hypersurfaces (effective divisors) are smooth. Poonen's<br />
work on Bertini theorems over finite fields answers this question when<br />
one considers effective divisors linearly equivalent to a multiple of<br />
a fixed ample divisor, which corresponds to choosing an ample ray<br />
through the origin in the Picard group of the variety. In this case<br />
the probability of smoothness is predicted by a simple heuristic<br />
assuming smoothness is independent at different points in the ambient<br />
space. In joint work with Erman, we consider this question for<br />
effective divisors along nef rays in certain surfaces. Here the<br />
simple heuristic of independence fails, but the answer can still be<br />
determined and follows from a richer heuristic that predicts at<br />
which points smoothness is independent and at which<br />
points it is dependent.<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== October 20 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Jie Ling''' (Madison)<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Arithmetic intersection on Toric schemes and resultants<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: Let ''K'' be a number field, ''O<sub>K</sub>'' its ring of integers. Consider ''n''&nbsp;+&nbsp;1 Laurent polynomials ''f<sub>i</sub>'' in ''n'' variables with ''O<sub>K</sub>'' coefficients. We assume that they have support in given polytopes &Delta;<sub>''i''</sub>. On one hand, we can associate a toric scheme ''X'' over ''O<sub>K</sub>'' to these polytopes, and consider the arithmetic intersection number of (''f''<sub>0</sub>,&nbsp;...,''f<sub>n</sub>'')<sub>''X''</sub> in ''X''. On the other hand, we have the mixed resultant Res(''f''<sub>0</sub>,&nbsp;...,''f<sub>n</sub>''). When the associated scheme is projective and smooth at the generic fiber and we assume ''f''<sub>0</sub>,&nbsp;...,''f<sub>n</sub>'' intersect properly, the arithmetic intersection number is given by the norm of the mixed resultant. This could be thought of as an arithmetic analog of Beinstein's theorem.<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== October 27 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Danny Neftin''' (Michigan)<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Arithmetic field relations and crossed product division algebras<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: A finite group ''G'' is called ''K''-admissible if there exists a Galois ''G''-extension ''L''/''K'' such that ''L'' is a maximal subfield of a division algebra with center ''K'' (i.e. if there exists a ''G''-crossed product ''K''-division algebra). Motivated by the works of Neukirch, Uchida, and Iwasawa, Sonn defined an arithmetic equivalence of number fields by admissibility and posed (1985) the following open problem: Do two number fields with the same admissible groups necessarily have the same degree over Q?<br />
Up until recently only positive results were known. We shall discuss equivalence by admissibility and construct two number fields that have the same odd order admissible groups, providing evidence to a negative answer to Sonn's problem.<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== November 3 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Zev Klagsburn''' (Madison)<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Selmer ranks of quadratic twists of elliptic curves<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: Given an elliptic curve ''E'' defined over a number field ''K'', we can ask what proportion of quadratic twists of ''E'' have 2-Selmer rank ''r'' for any non-negative integer ''r''. The Delaunay heuristics combined with work of Dokchitser and Dokchitser suggested a conjecture for this distribution that was verified by work of Heath-Brown, Swinnerton-Dyer, and Kane for elliptic curves over the rationals with ''E''('''Q''')[2] = '''Z'''/2&nbsp;&times;&nbsp;'''Z'''/2. We present new results for elliptic curves with ''E''(''K'')[2]&nbsp;=&nbsp;0 and with ''E''(''K'')[2]&nbsp;=&nbsp;'''Z'''/2. I will present joint work of Mazur, Rubin, and myself supporting the conjecture for curves with ''E''(''K'')[2]&nbsp;=&nbsp;0. Additionally, I will present some new results of my own for curves with ''E''(''K'')[2]&nbsp;=&nbsp;'''Z'''/2, including some surprising results that conflict with the conjecture.<br />
|} <br />
</center><br />
<br />
<br><br />
--><br />
<br />
== March 29 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''David P. Roberts''' (U. Minnesota Morris)<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: tba<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: tba<br />
<br />
|} <br />
</center><br />
<br />
<br />
<br><br />
<br />
== April 12 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Chenyan Wu''' (Minnesota)<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: tba<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: tba<br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== April 19 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Robert Guralnick''' (U. Southern California)<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: A variant of Burnside and Galois representations which are automorphic<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: Wiles, Taylor, Harris and others used the notion of a big<br />
representation of a finite<br />
group to show that certain representations are automorphic. Jack Thorne<br />
recently observed<br />
that one could weaken this notion of bigness to get the same conclusions. He<br />
called this property adequate. An absolutely irreducible representation ''V''<br />
of a finite group ''G'' in characteristic ''p'' is called adequate if ''G'' has<br />
no ''p''-quotients, the dimension<br />
of ''V'' is prime to ''p'', ''V'' has non-trivial self extensions and End(''V'') is<br />
generated by the linear<br />
span of the elements of order prime to ''p'' in ''G''. If ''G'' has order<br />
prime to ''p'', all of these conditions<br />
hold&mdash;the last condition is sometimes called Burnside's Lemma. We<br />
will discuss a recent<br />
result of Guralnick, Herzig, Taylor and Thorne showing that if ''p'' ><br />
2 dim ''V'' + 2, then<br />
any absolutely irreducible representation is adequate. We will also<br />
discuss some examples<br />
showing that the span of the ''p'''-elements in End(''V'') need not be all of End(''V'').<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== April 26 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Frank Thorne''' (U. South Carolina)<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: tba<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
<br />
Abstract: tba<br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== May 3 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Alina Cojocaru''' (U. Illinois at Chicago)<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: tba<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: tba<br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== May 10 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Samit Dasgupta''' (UC Santa Cruz)<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: tba<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: tba <br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== Organizer contact information ==<br />
<br />
[http://www.math.wisc.edu/~shamgar/ Shamgar Gurevich]<br />
<br />
[http://www.math.wisc.edu/~rharron/ Robert Harron]<br />
<br />
[http://www.math.wisc.edu/~klagsbru/ Zev Klagsbrun]<br />
<br />
[http://www.math.wisc.edu/~mmwood/ Melanie Matchett Wood]<br />
<br><br />
<br />
<br />
<br />
----<br />
Return to the [[NTS|Number Theory Seminar Page]]<br />
<br />
Return to the [[Algebra|Algebra Group Page]]</div>Klagsbruhttps://wiki.math.wisc.edu/index.php?title=NTS_Spring_2012/Abstracts&diff=3580NTS Spring 2012/Abstracts2012-02-27T20:30:46Z<p>Klagsbru: </p>
<hr />
<div>== February 2 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Evan Dummit''' (Madison)<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Kakeya sets over non-archimedean local rings<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: In a forthcoming paper with Marci Habliscek, we constructed a Kakeya set over the formal power series ring '''F'''<sub>''q''</sub><nowiki>[[</nowiki>''t''&thinsp;<nowiki>]]</nowiki>, answering a question posed by Ellenberg, Oberlin, and Tao. My talk will be devoted to explaining some of the older history of the Kakeya problem in analysis and the newer history of the Kakeya problem in combinatorics, including my joint work with Marci. In particular, I will give Dvir's solution of the Kakeya problem over finite fields, and explain the problem's extension to other classes of rings. <br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== February 16 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Tonghai Yang''' (Madison)<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: A little linear algebra on CM abelian surfaces<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: In this talk, I will discuss an interesting Hermitian form structure on the space of ''special endormorphisms'' of a CM abelian surface, and how to use it make a moduli problem and prove an arithmetic Siegel–Weil formula over a real quadratic field. This is a joint work with Ben Howard.<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== February 23 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Christelle Vincent''' (Madison)<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Drinfeld modular forms<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: We will begin by introducing the Drinfeld setting, and in particular Drinfeld modular forms and their connection to the geometry of Drinfeld modular curves. We will then present some results about Drinfeld modular forms that we obtained in the process of computing certain geometric points on Drinfeld modular curves. More precisely, we will talk about Drinfeld modular forms modulo ''P'', for ''P'' a prime ideal in '''F'''<sub>''q''</sub>[''T''&thinsp;], and about Drinfeld quasi-modular forms.<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== March 1 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Shamgar Gurevich''' (Madison)<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Computing the Matched Filter in Linear Time<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: In the digital radar problem we design a function (waveform) S(t) in the Hilbert space H&nbsp;=&nbsp;C('''Z'''/''p'') of complex valued functions on '''Z'''/''p''&nbsp;=&nbsp;{0,&nbsp;...,&nbsp;''p''&nbsp;&minus;&nbsp;1}, the integers modulo a prime number ''p''&nbsp;≫&nbsp;0. We transmit the function S(t) using the radar to the object that we want to detect. The wave S(t) hits the object, and is reflected back via the echo wave R(t) in H, which has the form <br />
<br />
R(t)&nbsp;=&nbsp;exp{2πiωt/''p''}⋅S(t+τ)&nbsp;+&nbsp;W(t), <br />
<br />
where W(t) in H is a white noise, and τ,&nbsp;ω in '''Z'''/''p'', encode the distance from, and velocity of, the object. <br />
<br />
Problem (digital radar problem) Extract τ,&nbsp;ω from R and S. <br />
<br />
In the lecture I first introduce the classical matched filter (MF) algorithm that suggests the 'traditional' way (using fast Fourier transform) to solve the digital radar problem in order of ''p''<sup>2</sup>⋅log(''p'') operations. I will then explain how to use techniques from group representation theory and arithmetic to design (construct) waveforms S(t) which enable us to introduce a fast matched filter (FMF) algorithm, that we call the "flag algorithm", which solves the digital radar problem in a much faster way of order of ''p''⋅log(''p'') operations. <br />
<br />
I will demonstrate additional applications to mobile communication, and global positioning system (GPS). <br />
<br />
This is a joint work with A. Fish (Math, Madison), R. Hadani (Math, Austin), A. Sayeed (Electrical Engineering, Madison), and O. Schwartz (Electrical Engineering and Computer Science, Berkeley). <br />
<br />
The lecture is suitable for general math/engineering audience.<br />
<br />
|} <br />
</center><br />
<br />
<br><br />
== March 8 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Zev Klagsbrun''' (Madison)<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Erdos-Kac Type Theorems<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: In it's most popular formulation, the Erdos-Kac Theorem gives a distribution on the number of distinct primes factors ("ω(n)") of the numbers up to "N". Variants of the Erdos-Kac Theorem yield distributions on additive functions in a surprising number of settings. This talk will outline the basics of the theory by focusing on some results of Granville and Soundararajan that allow one to easily prove Erdos-Kac type results for a variety of problems as well as present a recent result of my own using the Granville and Soundararajan framework.<br />
<br />
The lecture is suitable for general math audience.<br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== March 15 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Yongqiang Zhao''' (Madison)<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: tba<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: tba<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== March 22 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Paul Terwilliger''' (Madison)<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: tba<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: tba<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
<!--== September 29 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Nigel Boston''' (Madison)<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Non-abelian Cohen-Lenstra heuristics.<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: In 1983, Cohen and Lenstra observed that the frequency with which a given abelian ''p''-group ''A'' (''p'' odd) arises as the ''p''-class group of an imaginary quadratic field ''K'' is apparently proportional to 1/|Aut(''A'')|. The Galois group of the maximal<br />
unramified ''p''-extension of ''K'' has abelianization ''A'' and one might then ask how frequently a given ''p''-group ''G'' arises. We<br />
develop a theory wherein this frequency is inversely proportional to the size of its automorphism group in a new category<br />
and then test this against computations. If time permits, I shall describe progress on the real quadratic case. This is<br />
joint work with Michael Bush and Farshid Hajir.<br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== October 6 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Zhiwei Yun''' (MIT)<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Exceptional Lie groups as motivic Galois groups<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: More than two decades ago, Serre asked the following<br />
question: can exceptional Lie groups be realized as the motivic Galois<br />
group of some motive over a number field? The question has been open<br />
for exceptional groups other than ''G''<sub>2</sub>. In this talk, I will show how<br />
to use geometric Langlands theory to give a uniform construction of<br />
motives with motivic Galois groups ''E''<sub>7</sub>, ''E''<sub>8</sub> and ''G''<sub>2</sub>, hence giving an<br />
affirmative answer to Serre's question in these cases.<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== October 13 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Melanie Matchett Wood''' (Madison)<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: The probability that a curve over a finite field is smooth<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
<br />
Abstract: Given a fixed variety over a finite field, we ask what<br />
proportion of hypersurfaces (effective divisors) are smooth. Poonen's<br />
work on Bertini theorems over finite fields answers this question when<br />
one considers effective divisors linearly equivalent to a multiple of<br />
a fixed ample divisor, which corresponds to choosing an ample ray<br />
through the origin in the Picard group of the variety. In this case<br />
the probability of smoothness is predicted by a simple heuristic<br />
assuming smoothness is independent at different points in the ambient<br />
space. In joint work with Erman, we consider this question for<br />
effective divisors along nef rays in certain surfaces. Here the<br />
simple heuristic of independence fails, but the answer can still be<br />
determined and follows from a richer heuristic that predicts at<br />
which points smoothness is independent and at which<br />
points it is dependent.<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== October 20 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Jie Ling''' (Madison)<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Arithmetic intersection on Toric schemes and resultants<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: Let ''K'' be a number field, ''O<sub>K</sub>'' its ring of integers. Consider ''n''&nbsp;+&nbsp;1 Laurent polynomials ''f<sub>i</sub>'' in ''n'' variables with ''O<sub>K</sub>'' coefficients. We assume that they have support in given polytopes &Delta;<sub>''i''</sub>. On one hand, we can associate a toric scheme ''X'' over ''O<sub>K</sub>'' to these polytopes, and consider the arithmetic intersection number of (''f''<sub>0</sub>,&nbsp;...,''f<sub>n</sub>'')<sub>''X''</sub> in ''X''. On the other hand, we have the mixed resultant Res(''f''<sub>0</sub>,&nbsp;...,''f<sub>n</sub>''). When the associated scheme is projective and smooth at the generic fiber and we assume ''f''<sub>0</sub>,&nbsp;...,''f<sub>n</sub>'' intersect properly, the arithmetic intersection number is given by the norm of the mixed resultant. This could be thought of as an arithmetic analog of Beinstein's theorem.<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== October 27 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Danny Neftin''' (Michigan)<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Arithmetic field relations and crossed product division algebras<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: A finite group ''G'' is called ''K''-admissible if there exists a Galois ''G''-extension ''L''/''K'' such that ''L'' is a maximal subfield of a division algebra with center ''K'' (i.e. if there exists a ''G''-crossed product ''K''-division algebra). Motivated by the works of Neukirch, Uchida, and Iwasawa, Sonn defined an arithmetic equivalence of number fields by admissibility and posed (1985) the following open problem: Do two number fields with the same admissible groups necessarily have the same degree over Q?<br />
Up until recently only positive results were known. We shall discuss equivalence by admissibility and construct two number fields that have the same odd order admissible groups, providing evidence to a negative answer to Sonn's problem.<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== November 3 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Zev Klagsburn''' (Madison)<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Selmer ranks of quadratic twists of elliptic curves<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: Given an elliptic curve ''E'' defined over a number field ''K'', we can ask what proportion of quadratic twists of ''E'' have 2-Selmer rank ''r'' for any non-negative integer ''r''. The Delaunay heuristics combined with work of Dokchitser and Dokchitser suggested a conjecture for this distribution that was verified by work of Heath-Brown, Swinnerton-Dyer, and Kane for elliptic curves over the rationals with ''E''('''Q''')[2] = '''Z'''/2&nbsp;&times;&nbsp;'''Z'''/2. We present new results for elliptic curves with ''E''(''K'')[2]&nbsp;=&nbsp;0 and with ''E''(''K'')[2]&nbsp;=&nbsp;'''Z'''/2. I will present joint work of Mazur, Rubin, and myself supporting the conjecture for curves with ''E''(''K'')[2]&nbsp;=&nbsp;0. Additionally, I will present some new results of my own for curves with ''E''(''K'')[2]&nbsp;=&nbsp;'''Z'''/2, including some surprising results that conflict with the conjecture.<br />
|} <br />
</center><br />
<br />
<br><br />
--><br />
<br />
== March 29 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''David P. Roberts''' (U. Minnesota Morris)<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: tba<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: tba<br />
<br />
|} <br />
</center><br />
<br />
<br />
<br><br />
<br />
== April 12 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Chenyan Wu''' (Minnesota)<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: tba<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: tba<br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== April 19 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Robert Guralnick''' (U. Southern California)<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: A variant of Burnside and Galois representations which are automorphic<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: Wiles, Taylor, Harris and others used the notion of a big<br />
representation of a finite<br />
group to show that certain representations are automorphic. Jack Thorne<br />
recently observed<br />
that one could weaken this notion of bigness to get the same conclusions. He<br />
called this property adequate. An absolutely irreducible representation ''V''<br />
of a finite group ''G'' in characteristic ''p'' is called adequate if ''G'' has<br />
no ''p''-quotients, the dimension<br />
of ''V'' is prime to ''p'', ''V'' has non-trivial self extensions and End(''V'') is<br />
generated by the linear<br />
span of the elements of order prime to ''p'' in ''G''. If ''G'' has order<br />
prime to ''p'', all of these conditions<br />
hold&mdash;the last condition is sometimes called Burnside's Lemma. We<br />
will discuss a recent<br />
result of Guralnick, Herzig, Taylor and Thorne showing that if ''p'' ><br />
2 dim ''V'' + 2, then<br />
any absolutely irreducible representation is adequate. We will also<br />
discuss some examples<br />
showing that the span of the ''p'''-elements in End(''V'') need not be all of End(''V'').<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== April 26 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Frank Thorne''' (U. South Carolina)<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: tba<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
<br />
Abstract: tba<br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== May 3 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Alina Cojocaru''' (U. Illinois at Chicago)<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: tba<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: tba<br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== May 10 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Samit Dasgupta''' (UC Santa Cruz)<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: tba<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: tba <br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== Organizer contact information ==<br />
<br />
[http://www.math.wisc.edu/~shamgar/ Shamgar Gurevich]<br />
<br />
[http://www.math.wisc.edu/~rharron/ Robert Harron]<br />
<br />
[http://www.math.wisc.edu/~klagsbru/ Zev Klagsbrun]<br />
<br />
[http://www.math.wisc.edu/~mmwood/ Melanie Matchett Wood]<br />
<br><br />
<br />
<br />
<br />
----<br />
Return to the [[NTS|Number Theory Seminar Page]]<br />
<br />
Return to the [[Algebra|Algebra Group Page]]</div>Klagsbruhttps://wiki.math.wisc.edu/index.php?title=NTS_Fall_2011/Abstracts&diff=2974NTS Fall 2011/Abstracts2011-10-31T16:06:10Z<p>Klagsbru: </p>
<hr />
<div>== September 8 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Alexander Fish''' (Madison)<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Solvability of Diophantine equations within dynamically defined subsets of N<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: Given a dynamical system, i.e. a compact metric space ''X'', a homeomorphism ''T'' (or just a continuous map) and a Borel probability measure on ''X'' which is preserved under the action of ''T'', the dynamically defined subset associated to a point ''x'' in ''X'' and an open set ''U'' in ''X'' is {''n'' | ''T<sup>&thinsp;n</sup>''(''x'') is in ''U''} which we call the set of return times of ''x'' in ''U''. We study combinatorial properties of sets of return times for certain types of dynamical systems for generic points ''x'' in ''X''. Among examples of such sets are normal sets which correspond to the system ''X''&nbsp;=&nbsp;[0,1], ''T''(''x'')&nbsp;=&nbsp;2''x''&nbsp;mod&nbsp;1, Lebesgue measure, ''U''&nbsp;=&nbsp;[0,&nbsp;1/2].<br />
We give a complete classification of linear Diophantine systems solvable within every normal set. The methods combine the probabilistic method together with the use of van der Corput's lemma. At the end of the talk we will discuss open problems<br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== September 15 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Chung Pang Mok''' (McMaster)<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Galois representation associated to cusp forms on GL<sub>2</sub> over CM fields<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: We generalize the work of Harris–Soudry–Taylor, and constructs<br />
the compatible system of 2-dimensional ''p''-adic Galois representations<br />
associated to a cuspidal automorphic representation of cohomological type<br />
on GL<sub>2</sub> over a CM field, whose central character satisfies an invariance<br />
condition. A local-global compatibility statement, up to<br />
semi-simplification, can also be proved in this setting. This work relies<br />
crucially on Arthur's results on lifting from the group GSp<sub>4</sub> to GL<sub>4</sub>.<br />
<br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== September 22 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Yifeng Liu''' (Columbia)<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Arithmetic inner product formula<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: I will introduce an arithmetic version of the classical Rallis' inner product<br />
for unitary groups, which generalizes the previous work by Kudla, Kudla–Rapoport–Yang and<br />
Bruinier–Yang. The arithmetic inner product formula, which is still a conjecture for<br />
higher rank, relates the canonical height of special cycles on certain Shimura varieties<br />
and the central derivatives of ''L''-functions.<br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== September 29 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Nigel Boston''' (Madison)<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Non-abelian Cohen-Lenstra heuristics.<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: In 1983, Cohen and Lenstra observed that the frequency with which a given abelian ''p''-group ''A'' (''p'' odd) arises as the ''p''-class group of an imaginary quadratic field ''K'' is apparently proportional to 1/|Aut(''A'')|. The Galois group of the maximal<br />
unramified ''p''-extension of ''K'' has abelianization ''A'' and one might then ask how frequently a given ''p''-group ''G'' arises. We<br />
develop a theory wherein this frequency is inversely proportional to the size of its automorphism group in a new category<br />
and then test this against computations. If time permits, I shall describe progress on the real quadratic case. This is<br />
joint work with Michael Bush and Farshid Hajir.<br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== October 6 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Zhiwei Yun''' (MIT)<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Exceptional Lie groups as motivic Galois groups<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: More than two decades ago, Serre asked the following<br />
question: can exceptional Lie groups be realized as the motivic Galois<br />
group of some motive over a number field? The question has been open<br />
for exceptional groups other than ''G''<sub>2</sub>. In this talk, I will show how<br />
to use geometric Langlands theory to give a uniform construction of<br />
motives with motivic Galois groups ''E''<sub>7</sub>, ''E''<sub>8</sub> and ''G''<sub>2</sub>, hence giving an<br />
affirmative answer to Serre's question in these cases.<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== October 13 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Melanie Matchett Wood''' (Madison)<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: The probability that a curve over a finite field is smooth<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
<br />
Abstract: Given a fixed variety over a finite field, we ask what<br />
proportion of hypersurfaces (effective divisors) are smooth. Poonen's<br />
work on Bertini theorems over finite fields answers this question when<br />
one considers effective divisors linearly equivalent to a multiple of<br />
a fixed ample divisor, which corresponds to choosing an ample ray<br />
through the origin in the Picard group of the variety. In this case<br />
the probability of smoothness is predicted by a simple heuristic<br />
assuming smoothness is independent at different points in the ambient<br />
space. In joint work with Erman, we consider this question for<br />
effective divisors along nef rays in certain surfaces. Here the<br />
simple heuristic of independence fails, but the answer can still be<br />
determined and follows from a richer heuristic that predicts at<br />
which points smoothness is independent and at which<br />
points it is dependent.<br />
<br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== October 20 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Jie Ling''' (Madison)<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Arithmetic intersection on Toric schemes and resultants<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: Let ''K'' be a number field, ''O<sub>K</sub>'' its ring of integers. Consider ''n''&nbsp;+&nbsp;1 Laurent polynomials ''f<sub>i</sub>'' in ''n'' variables with ''O<sub>K</sub>'' coefficients. We assume that they have support in given polytopes &Delta;<sub>''i''</sub>. On one hand, we can associate a toric scheme ''X'' over ''O<sub>K</sub>'' to these polytopes, and consider the arithmetic intersection number of (''f''<sub>0</sub>,&nbsp;...,''f<sub>n</sub>'')<sub>''X''</sub> in ''X''. On the other hand, we have the mixed resultant Res(''f''<sub>0</sub>,&nbsp;...,''f<sub>n</sub>''). When the associated scheme is projective and smooth at the generic fiber and we assume ''f''<sub>0</sub>,&nbsp;...,''f<sub>n</sub>'' intersect properly, the arithmetic intersection number is given by the norm of the mixed resultant. This could be thought of as an arithmetic analog of Beinstein's theorem.<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== October 27 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Danny Neftin''' (Michigan)<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Arithmetic field relations and crossed product division algebras<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: A finite group ''G'' is called ''K''-admissible if there exists a Galois ''G''-extension ''L''/''K'' such that ''L'' is a maximal subfield of a division algebra with center ''K'' (i.e. if there exists a ''G''-crossed product ''K''-division algebra). Motivated by the works of Neukirch, Uchida, and Iwasawa, Sonn defined an arithmetic equivalence of number fields by admissibility and posed (1985) the following open problem: Do two number fields with the same admissible groups necessarily have the same degree over Q?<br />
Up until recently only positive results were known. We shall discuss equivalence by admissibility and construct two number fields that have the same odd order admissible groups, providing evidence to a negative answer to Sonn's problem.<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== November 3 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Zev Klagsburn''' (Madison)<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Selmer Ranks of Quadratic Twists of Elliptic Curves<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: Given an elliptic curve ''E'' defined over a number field ''K'', we can ask what proportion of quadratic twists of ''E'' have 2-Selmer rank ''r'' for any non-negative integer ''r''. The Delaunay heuristics combined with work of Dokchitser and Dokchitser suggested a conjecture for this distribution that was verified by work of Heath-Brown, Swinnerton-Dyer, and Kane for elliptic curves over the rationals with ''E(Q)[2] = Z/2 x Z/2''. We present new results for elliptic curves with ''E(K)[2] = 0'' and with ''E(K)[2] = Z/2''. I will present joint work of Mazur, Rubin, and myself supporting the conjecture for curves with ''E(K)[2] = 0'' Additionally, I will present some new results of my own for curves with ''E(K)[2] =Z/2'', including some surprising results that conflict with the conjecture.<br />
<br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== November 10 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Luanlei Zhao''' (Madison)<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: tba<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: tba<br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== November 17 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Robert Harron''' (Madison)<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: tba<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: tba<br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== December 1 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Andrei Calderaru''' (Madison)<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: tba<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: tba<br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== December 8 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Xinwen Zhu''' (Harvard)<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: tba<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: tba<br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== December 15 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Shamgar Gurevich''' (Madison)<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Quadratic reciprocity and the sign of the Gauss sum via the finite Weil representation<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: tba<br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== Organizer contact information ==<br />
<br />
[http://www.math.wisc.edu/~shamgar/ Shamgar Gurevich]<br />
<br />
[http://www.math.wisc.edu/~rharron/ Robert Harron]<br />
<br />
[http://www.math.wisc.edu/~klagsbru/ Zev Klagsbrun]<br />
<br />
[http://www.math.wisc.edu/~mmwood/ Melanie Matchett Wood]<br />
<br><br />
<br />
<br />
<br />
----<br />
Return to the [[NTS|Number Theory Seminar Page]]<br />
<br />
Return to the [[Algebra|Algebra Group Page]]</div>Klagsbruhttps://wiki.math.wisc.edu/index.php?title=NTS_Fall_2011/Abstracts&diff=2973NTS Fall 2011/Abstracts2011-10-31T16:05:41Z<p>Klagsbru: </p>
<hr />
<div>== September 8 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Alexander Fish''' (Madison)<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Solvability of Diophantine equations within dynamically defined subsets of N<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: Given a dynamical system, i.e. a compact metric space ''X'', a homeomorphism ''T'' (or just a continuous map) and a Borel probability measure on ''X'' which is preserved under the action of ''T'', the dynamically defined subset associated to a point ''x'' in ''X'' and an open set ''U'' in ''X'' is {''n'' | ''T<sup>&thinsp;n</sup>''(''x'') is in ''U''} which we call the set of return times of ''x'' in ''U''. We study combinatorial properties of sets of return times for certain types of dynamical systems for generic points ''x'' in ''X''. Among examples of such sets are normal sets which correspond to the system ''X''&nbsp;=&nbsp;[0,1], ''T''(''x'')&nbsp;=&nbsp;2''x''&nbsp;mod&nbsp;1, Lebesgue measure, ''U''&nbsp;=&nbsp;[0,&nbsp;1/2].<br />
We give a complete classification of linear Diophantine systems solvable within every normal set. The methods combine the probabilistic method together with the use of van der Corput's lemma. At the end of the talk we will discuss open problems<br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== September 15 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Chung Pang Mok''' (McMaster)<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Galois representation associated to cusp forms on GL<sub>2</sub> over CM fields<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: We generalize the work of Harris–Soudry–Taylor, and constructs<br />
the compatible system of 2-dimensional ''p''-adic Galois representations<br />
associated to a cuspidal automorphic representation of cohomological type<br />
on GL<sub>2</sub> over a CM field, whose central character satisfies an invariance<br />
condition. A local-global compatibility statement, up to<br />
semi-simplification, can also be proved in this setting. This work relies<br />
crucially on Arthur's results on lifting from the group GSp<sub>4</sub> to GL<sub>4</sub>.<br />
<br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== September 22 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Yifeng Liu''' (Columbia)<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Arithmetic inner product formula<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: I will introduce an arithmetic version of the classical Rallis' inner product<br />
for unitary groups, which generalizes the previous work by Kudla, Kudla–Rapoport–Yang and<br />
Bruinier–Yang. The arithmetic inner product formula, which is still a conjecture for<br />
higher rank, relates the canonical height of special cycles on certain Shimura varieties<br />
and the central derivatives of ''L''-functions.<br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== September 29 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Nigel Boston''' (Madison)<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Non-abelian Cohen-Lenstra heuristics.<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: In 1983, Cohen and Lenstra observed that the frequency with which a given abelian ''p''-group ''A'' (''p'' odd) arises as the ''p''-class group of an imaginary quadratic field ''K'' is apparently proportional to 1/|Aut(''A'')|. The Galois group of the maximal<br />
unramified ''p''-extension of ''K'' has abelianization ''A'' and one might then ask how frequently a given ''p''-group ''G'' arises. We<br />
develop a theory wherein this frequency is inversely proportional to the size of its automorphism group in a new category<br />
and then test this against computations. If time permits, I shall describe progress on the real quadratic case. This is<br />
joint work with Michael Bush and Farshid Hajir.<br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== October 6 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Zhiwei Yun''' (MIT)<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Exceptional Lie groups as motivic Galois groups<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: More than two decades ago, Serre asked the following<br />
question: can exceptional Lie groups be realized as the motivic Galois<br />
group of some motive over a number field? The question has been open<br />
for exceptional groups other than ''G''<sub>2</sub>. In this talk, I will show how<br />
to use geometric Langlands theory to give a uniform construction of<br />
motives with motivic Galois groups ''E''<sub>7</sub>, ''E''<sub>8</sub> and ''G''<sub>2</sub>, hence giving an<br />
affirmative answer to Serre's question in these cases.<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== October 13 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Melanie Matchett Wood''' (Madison)<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: The probability that a curve over a finite field is smooth<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
<br />
Abstract: Given a fixed variety over a finite field, we ask what<br />
proportion of hypersurfaces (effective divisors) are smooth. Poonen's<br />
work on Bertini theorems over finite fields answers this question when<br />
one considers effective divisors linearly equivalent to a multiple of<br />
a fixed ample divisor, which corresponds to choosing an ample ray<br />
through the origin in the Picard group of the variety. In this case<br />
the probability of smoothness is predicted by a simple heuristic<br />
assuming smoothness is independent at different points in the ambient<br />
space. In joint work with Erman, we consider this question for<br />
effective divisors along nef rays in certain surfaces. Here the<br />
simple heuristic of independence fails, but the answer can still be<br />
determined and follows from a richer heuristic that predicts at<br />
which points smoothness is independent and at which<br />
points it is dependent.<br />
<br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== October 20 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Jie Ling''' (Madison)<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Arithmetic intersection on Toric schemes and resultants<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: Let ''K'' be a number field, ''O<sub>K</sub>'' its ring of integers. Consider ''n''&nbsp;+&nbsp;1 Laurent polynomials ''f<sub>i</sub>'' in ''n'' variables with ''O<sub>K</sub>'' coefficients. We assume that they have support in given polytopes &Delta;<sub>''i''</sub>. On one hand, we can associate a toric scheme ''X'' over ''O<sub>K</sub>'' to these polytopes, and consider the arithmetic intersection number of (''f''<sub>0</sub>,&nbsp;...,''f<sub>n</sub>'')<sub>''X''</sub> in ''X''. On the other hand, we have the mixed resultant Res(''f''<sub>0</sub>,&nbsp;...,''f<sub>n</sub>''). When the associated scheme is projective and smooth at the generic fiber and we assume ''f''<sub>0</sub>,&nbsp;...,''f<sub>n</sub>'' intersect properly, the arithmetic intersection number is given by the norm of the mixed resultant. This could be thought of as an arithmetic analog of Beinstein's theorem.<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== October 27 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Danny Neftin''' (Michigan)<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Arithmetic field relations and crossed product division algebras<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: A finite group ''G'' is called ''K''-admissible if there exists a Galois ''G''-extension ''L''/''K'' such that ''L'' is a maximal subfield of a division algebra with center ''K'' (i.e. if there exists a ''G''-crossed product ''K''-division algebra). Motivated by the works of Neukirch, Uchida, and Iwasawa, Sonn defined an arithmetic equivalence of number fields by admissibility and posed (1985) the following open problem: Do two number fields with the same admissible groups necessarily have the same degree over Q?<br />
Up until recently only positive results were known. We shall discuss equivalence by admissibility and construct two number fields that have the same odd order admissible groups, providing evidence to a negative answer to Sonn's problem.<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== November 3 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Zev Klagsburn''' (Madison)<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Selmer Ranks of Quadratic Twists of Elliptic Curves<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: Given an elliptic curve ''E'' defined over a number field ''K'', we can ask what proportion of quadratic twists of ''E'' have 2-Selmer rank ''r'' for any non-negative integer ''r''. The Delaunay heuristics combined with work of Dokchitser and Dokchitser suggested a conjecture for this distribution that was verified by work of Heath-Brown, Swinnerton-Dyer, and Kane for elliptic curves over the rationals with ''E(\mathbb{Q})[2] = Z/2 x Z/2''. We present new results for elliptic curves with ''E(K)[2] = 0'' and with ''E(K)[2] = Z/2''. I will present joint work of Mazur, Rubin, and myself supporting the conjecture for curves with ''E(K)[2] = 0'' Additionally, I will present some new results of my own for curves with ''E(K)[2] =Z/2'', including some surprising results that conflict with the conjecture.<br />
<br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== November 10 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Luanlei Zhao''' (Madison)<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: tba<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: tba<br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== November 17 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Robert Harron''' (Madison)<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: tba<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: tba<br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== December 1 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Andrei Calderaru''' (Madison)<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: tba<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: tba<br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== December 8 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Xinwen Zhu''' (Harvard)<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: tba<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: tba<br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== December 15 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Shamgar Gurevich''' (Madison)<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Quadratic reciprocity and the sign of the Gauss sum via the finite Weil representation<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: tba<br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== Organizer contact information ==<br />
<br />
[http://www.math.wisc.edu/~shamgar/ Shamgar Gurevich]<br />
<br />
[http://www.math.wisc.edu/~rharron/ Robert Harron]<br />
<br />
[http://www.math.wisc.edu/~klagsbru/ Zev Klagsbrun]<br />
<br />
[http://www.math.wisc.edu/~mmwood/ Melanie Matchett Wood]<br />
<br><br />
<br />
<br />
<br />
----<br />
Return to the [[NTS|Number Theory Seminar Page]]<br />
<br />
Return to the [[Algebra|Algebra Group Page]]</div>Klagsbruhttps://wiki.math.wisc.edu/index.php?title=NTS_Fall_2011/Abstracts&diff=2972NTS Fall 2011/Abstracts2011-10-31T16:05:19Z<p>Klagsbru: </p>
<hr />
<div>== September 8 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Alexander Fish''' (Madison)<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Solvability of Diophantine equations within dynamically defined subsets of N<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: Given a dynamical system, i.e. a compact metric space ''X'', a homeomorphism ''T'' (or just a continuous map) and a Borel probability measure on ''X'' which is preserved under the action of ''T'', the dynamically defined subset associated to a point ''x'' in ''X'' and an open set ''U'' in ''X'' is {''n'' | ''T<sup>&thinsp;n</sup>''(''x'') is in ''U''} which we call the set of return times of ''x'' in ''U''. We study combinatorial properties of sets of return times for certain types of dynamical systems for generic points ''x'' in ''X''. Among examples of such sets are normal sets which correspond to the system ''X''&nbsp;=&nbsp;[0,1], ''T''(''x'')&nbsp;=&nbsp;2''x''&nbsp;mod&nbsp;1, Lebesgue measure, ''U''&nbsp;=&nbsp;[0,&nbsp;1/2].<br />
We give a complete classification of linear Diophantine systems solvable within every normal set. The methods combine the probabilistic method together with the use of van der Corput's lemma. At the end of the talk we will discuss open problems<br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== September 15 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Chung Pang Mok''' (McMaster)<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Galois representation associated to cusp forms on GL<sub>2</sub> over CM fields<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: We generalize the work of Harris–Soudry–Taylor, and constructs<br />
the compatible system of 2-dimensional ''p''-adic Galois representations<br />
associated to a cuspidal automorphic representation of cohomological type<br />
on GL<sub>2</sub> over a CM field, whose central character satisfies an invariance<br />
condition. A local-global compatibility statement, up to<br />
semi-simplification, can also be proved in this setting. This work relies<br />
crucially on Arthur's results on lifting from the group GSp<sub>4</sub> to GL<sub>4</sub>.<br />
<br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== September 22 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Yifeng Liu''' (Columbia)<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Arithmetic inner product formula<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: I will introduce an arithmetic version of the classical Rallis' inner product<br />
for unitary groups, which generalizes the previous work by Kudla, Kudla–Rapoport–Yang and<br />
Bruinier–Yang. The arithmetic inner product formula, which is still a conjecture for<br />
higher rank, relates the canonical height of special cycles on certain Shimura varieties<br />
and the central derivatives of ''L''-functions.<br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== September 29 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Nigel Boston''' (Madison)<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Non-abelian Cohen-Lenstra heuristics.<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: In 1983, Cohen and Lenstra observed that the frequency with which a given abelian ''p''-group ''A'' (''p'' odd) arises as the ''p''-class group of an imaginary quadratic field ''K'' is apparently proportional to 1/|Aut(''A'')|. The Galois group of the maximal<br />
unramified ''p''-extension of ''K'' has abelianization ''A'' and one might then ask how frequently a given ''p''-group ''G'' arises. We<br />
develop a theory wherein this frequency is inversely proportional to the size of its automorphism group in a new category<br />
and then test this against computations. If time permits, I shall describe progress on the real quadratic case. This is<br />
joint work with Michael Bush and Farshid Hajir.<br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== October 6 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Zhiwei Yun''' (MIT)<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Exceptional Lie groups as motivic Galois groups<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: More than two decades ago, Serre asked the following<br />
question: can exceptional Lie groups be realized as the motivic Galois<br />
group of some motive over a number field? The question has been open<br />
for exceptional groups other than ''G''<sub>2</sub>. In this talk, I will show how<br />
to use geometric Langlands theory to give a uniform construction of<br />
motives with motivic Galois groups ''E''<sub>7</sub>, ''E''<sub>8</sub> and ''G''<sub>2</sub>, hence giving an<br />
affirmative answer to Serre's question in these cases.<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== October 13 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Melanie Matchett Wood''' (Madison)<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: The probability that a curve over a finite field is smooth<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
<br />
Abstract: Given a fixed variety over a finite field, we ask what<br />
proportion of hypersurfaces (effective divisors) are smooth. Poonen's<br />
work on Bertini theorems over finite fields answers this question when<br />
one considers effective divisors linearly equivalent to a multiple of<br />
a fixed ample divisor, which corresponds to choosing an ample ray<br />
through the origin in the Picard group of the variety. In this case<br />
the probability of smoothness is predicted by a simple heuristic<br />
assuming smoothness is independent at different points in the ambient<br />
space. In joint work with Erman, we consider this question for<br />
effective divisors along nef rays in certain surfaces. Here the<br />
simple heuristic of independence fails, but the answer can still be<br />
determined and follows from a richer heuristic that predicts at<br />
which points smoothness is independent and at which<br />
points it is dependent.<br />
<br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== October 20 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Jie Ling''' (Madison)<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Arithmetic intersection on Toric schemes and resultants<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: Let ''K'' be a number field, ''O<sub>K</sub>'' its ring of integers. Consider ''n''&nbsp;+&nbsp;1 Laurent polynomials ''f<sub>i</sub>'' in ''n'' variables with ''O<sub>K</sub>'' coefficients. We assume that they have support in given polytopes &Delta;<sub>''i''</sub>. On one hand, we can associate a toric scheme ''X'' over ''O<sub>K</sub>'' to these polytopes, and consider the arithmetic intersection number of (''f''<sub>0</sub>,&nbsp;...,''f<sub>n</sub>'')<sub>''X''</sub> in ''X''. On the other hand, we have the mixed resultant Res(''f''<sub>0</sub>,&nbsp;...,''f<sub>n</sub>''). When the associated scheme is projective and smooth at the generic fiber and we assume ''f''<sub>0</sub>,&nbsp;...,''f<sub>n</sub>'' intersect properly, the arithmetic intersection number is given by the norm of the mixed resultant. This could be thought of as an arithmetic analog of Beinstein's theorem.<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== October 27 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Danny Neftin''' (Michigan)<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Arithmetic field relations and crossed product division algebras<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: A finite group ''G'' is called ''K''-admissible if there exists a Galois ''G''-extension ''L''/''K'' such that ''L'' is a maximal subfield of a division algebra with center ''K'' (i.e. if there exists a ''G''-crossed product ''K''-division algebra). Motivated by the works of Neukirch, Uchida, and Iwasawa, Sonn defined an arithmetic equivalence of number fields by admissibility and posed (1985) the following open problem: Do two number fields with the same admissible groups necessarily have the same degree over Q?<br />
Up until recently only positive results were known. We shall discuss equivalence by admissibility and construct two number fields that have the same odd order admissible groups, providing evidence to a negative answer to Sonn's problem.<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== November 3 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Zev Klagsburn''' (Madison)<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Selmer Ranks of Quadratic Twists of Elliptic Curves<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: Given an elliptic curve ''E'' defined over a number field ''K'', we can ask what proportion of quadratic twists of "K" have 2-Selmer rank ''r'' for any non-negative integer ''r''. The Delaunay heuristics combined with work of Dokchitser and Dokchitser suggested a conjecture for this distribution that was verified by work of Heath-Brown, Swinnerton-Dyer, and Kane for elliptic curves over the rationals with ''E(\mathbb{Q})[2] = Z/2 x Z/2''. We present new results for elliptic curves with ''E(K)[2] = 0'' and with ''E(K)[2] = Z/2''. I will present joint work of Mazur, Rubin, and myself supporting the conjecture for curves with ''E(K)[2] = 0'' Additionally, I will present some new results of my own for curves with ''E(K)[2] =Z/2'', including some surprising results that conflict with the conjecture.<br />
<br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== November 10 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Luanlei Zhao''' (Madison)<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: tba<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: tba<br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== November 17 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Robert Harron''' (Madison)<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: tba<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: tba<br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== December 1 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Andrei Calderaru''' (Madison)<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: tba<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: tba<br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== December 8 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Xinwen Zhu''' (Harvard)<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: tba<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: tba<br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== December 15 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Shamgar Gurevich''' (Madison)<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Quadratic reciprocity and the sign of the Gauss sum via the finite Weil representation<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: tba<br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== Organizer contact information ==<br />
<br />
[http://www.math.wisc.edu/~shamgar/ Shamgar Gurevich]<br />
<br />
[http://www.math.wisc.edu/~rharron/ Robert Harron]<br />
<br />
[http://www.math.wisc.edu/~klagsbru/ Zev Klagsbrun]<br />
<br />
[http://www.math.wisc.edu/~mmwood/ Melanie Matchett Wood]<br />
<br><br />
<br />
<br />
<br />
----<br />
Return to the [[NTS|Number Theory Seminar Page]]<br />
<br />
Return to the [[Algebra|Algebra Group Page]]</div>Klagsbruhttps://wiki.math.wisc.edu/index.php?title=NTS_Fall_2011/Abstracts&diff=2971NTS Fall 2011/Abstracts2011-10-31T16:04:53Z<p>Klagsbru: </p>
<hr />
<div>== September 8 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Alexander Fish''' (Madison)<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Solvability of Diophantine equations within dynamically defined subsets of N<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: Given a dynamical system, i.e. a compact metric space ''X'', a homeomorphism ''T'' (or just a continuous map) and a Borel probability measure on ''X'' which is preserved under the action of ''T'', the dynamically defined subset associated to a point ''x'' in ''X'' and an open set ''U'' in ''X'' is {''n'' | ''T<sup>&thinsp;n</sup>''(''x'') is in ''U''} which we call the set of return times of ''x'' in ''U''. We study combinatorial properties of sets of return times for certain types of dynamical systems for generic points ''x'' in ''X''. Among examples of such sets are normal sets which correspond to the system ''X''&nbsp;=&nbsp;[0,1], ''T''(''x'')&nbsp;=&nbsp;2''x''&nbsp;mod&nbsp;1, Lebesgue measure, ''U''&nbsp;=&nbsp;[0,&nbsp;1/2].<br />
We give a complete classification of linear Diophantine systems solvable within every normal set. The methods combine the probabilistic method together with the use of van der Corput's lemma. At the end of the talk we will discuss open problems<br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== September 15 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Chung Pang Mok''' (McMaster)<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Galois representation associated to cusp forms on GL<sub>2</sub> over CM fields<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: We generalize the work of Harris–Soudry–Taylor, and constructs<br />
the compatible system of 2-dimensional ''p''-adic Galois representations<br />
associated to a cuspidal automorphic representation of cohomological type<br />
on GL<sub>2</sub> over a CM field, whose central character satisfies an invariance<br />
condition. A local-global compatibility statement, up to<br />
semi-simplification, can also be proved in this setting. This work relies<br />
crucially on Arthur's results on lifting from the group GSp<sub>4</sub> to GL<sub>4</sub>.<br />
<br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== September 22 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Yifeng Liu''' (Columbia)<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Arithmetic inner product formula<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: I will introduce an arithmetic version of the classical Rallis' inner product<br />
for unitary groups, which generalizes the previous work by Kudla, Kudla–Rapoport–Yang and<br />
Bruinier–Yang. The arithmetic inner product formula, which is still a conjecture for<br />
higher rank, relates the canonical height of special cycles on certain Shimura varieties<br />
and the central derivatives of ''L''-functions.<br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== September 29 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Nigel Boston''' (Madison)<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Non-abelian Cohen-Lenstra heuristics.<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: In 1983, Cohen and Lenstra observed that the frequency with which a given abelian ''p''-group ''A'' (''p'' odd) arises as the ''p''-class group of an imaginary quadratic field ''K'' is apparently proportional to 1/|Aut(''A'')|. The Galois group of the maximal<br />
unramified ''p''-extension of ''K'' has abelianization ''A'' and one might then ask how frequently a given ''p''-group ''G'' arises. We<br />
develop a theory wherein this frequency is inversely proportional to the size of its automorphism group in a new category<br />
and then test this against computations. If time permits, I shall describe progress on the real quadratic case. This is<br />
joint work with Michael Bush and Farshid Hajir.<br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== October 6 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Zhiwei Yun''' (MIT)<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Exceptional Lie groups as motivic Galois groups<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: More than two decades ago, Serre asked the following<br />
question: can exceptional Lie groups be realized as the motivic Galois<br />
group of some motive over a number field? The question has been open<br />
for exceptional groups other than ''G''<sub>2</sub>. In this talk, I will show how<br />
to use geometric Langlands theory to give a uniform construction of<br />
motives with motivic Galois groups ''E''<sub>7</sub>, ''E''<sub>8</sub> and ''G''<sub>2</sub>, hence giving an<br />
affirmative answer to Serre's question in these cases.<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== October 13 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Melanie Matchett Wood''' (Madison)<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: The probability that a curve over a finite field is smooth<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
<br />
Abstract: Given a fixed variety over a finite field, we ask what<br />
proportion of hypersurfaces (effective divisors) are smooth. Poonen's<br />
work on Bertini theorems over finite fields answers this question when<br />
one considers effective divisors linearly equivalent to a multiple of<br />
a fixed ample divisor, which corresponds to choosing an ample ray<br />
through the origin in the Picard group of the variety. In this case<br />
the probability of smoothness is predicted by a simple heuristic<br />
assuming smoothness is independent at different points in the ambient<br />
space. In joint work with Erman, we consider this question for<br />
effective divisors along nef rays in certain surfaces. Here the<br />
simple heuristic of independence fails, but the answer can still be<br />
determined and follows from a richer heuristic that predicts at<br />
which points smoothness is independent and at which<br />
points it is dependent.<br />
<br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== October 20 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Jie Ling''' (Madison)<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Arithmetic intersection on Toric schemes and resultants<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: Let ''K'' be a number field, ''O<sub>K</sub>'' its ring of integers. Consider ''n''&nbsp;+&nbsp;1 Laurent polynomials ''f<sub>i</sub>'' in ''n'' variables with ''O<sub>K</sub>'' coefficients. We assume that they have support in given polytopes &Delta;<sub>''i''</sub>. On one hand, we can associate a toric scheme ''X'' over ''O<sub>K</sub>'' to these polytopes, and consider the arithmetic intersection number of (''f''<sub>0</sub>,&nbsp;...,''f<sub>n</sub>'')<sub>''X''</sub> in ''X''. On the other hand, we have the mixed resultant Res(''f''<sub>0</sub>,&nbsp;...,''f<sub>n</sub>''). When the associated scheme is projective and smooth at the generic fiber and we assume ''f''<sub>0</sub>,&nbsp;...,''f<sub>n</sub>'' intersect properly, the arithmetic intersection number is given by the norm of the mixed resultant. This could be thought of as an arithmetic analog of Beinstein's theorem.<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== October 27 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Danny Neftin''' (Michigan)<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Arithmetic field relations and crossed product division algebras<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: A finite group ''G'' is called ''K''-admissible if there exists a Galois ''G''-extension ''L''/''K'' such that ''L'' is a maximal subfield of a division algebra with center ''K'' (i.e. if there exists a ''G''-crossed product ''K''-division algebra). Motivated by the works of Neukirch, Uchida, and Iwasawa, Sonn defined an arithmetic equivalence of number fields by admissibility and posed (1985) the following open problem: Do two number fields with the same admissible groups necessarily have the same degree over Q?<br />
Up until recently only positive results were known. We shall discuss equivalence by admissibility and construct two number fields that have the same odd order admissible groups, providing evidence to a negative answer to Sonn's problem.<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== November 3 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Zev Klagsburn''' (Madison)<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Selmer Ranks of Quadratic Twists of Elliptic Curves<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: Given an elliptic curve "E" defined over a number field ''K'', we can ask what proportion of quadratic twists of "K" have 2-Selmer rank ''r'' for any non-negative integer ''r''. The Delaunay heuristics combined with work of Dokchitser and Dokchitser suggested a conjecture for this distribution that was verified by work of Heath-Brown, Swinnerton-Dyer, and Kane for elliptic curves over the rationals with ''E(\mathbb{Q})[2] = Z/2 x Z/2''. We present new results for elliptic curves with ''E(K)[2] = 0'' and with ''E(K)[2] = Z/2''. I will present joint work of Mazur, Rubin, and myself supporting the conjecture for curves with ''E(K)[2] = 0'' Additionally, I will present some new results of my own for curves with ''E(K)[2] =Z/2'', including some surprising results that conflict with the conjecture.<br />
<br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== November 10 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Luanlei Zhao''' (Madison)<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: tba<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: tba<br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== November 17 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Robert Harron''' (Madison)<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: tba<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: tba<br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== December 1 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Andrei Calderaru''' (Madison)<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: tba<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: tba<br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== December 8 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Xinwen Zhu''' (Harvard)<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: tba<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: tba<br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== December 15 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Shamgar Gurevich''' (Madison)<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Quadratic reciprocity and the sign of the Gauss sum via the finite Weil representation<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: tba<br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== Organizer contact information ==<br />
<br />
[http://www.math.wisc.edu/~shamgar/ Shamgar Gurevich]<br />
<br />
[http://www.math.wisc.edu/~rharron/ Robert Harron]<br />
<br />
[http://www.math.wisc.edu/~klagsbru/ Zev Klagsbrun]<br />
<br />
[http://www.math.wisc.edu/~mmwood/ Melanie Matchett Wood]<br />
<br><br />
<br />
<br />
<br />
----<br />
Return to the [[NTS|Number Theory Seminar Page]]<br />
<br />
Return to the [[Algebra|Algebra Group Page]]</div>Klagsbruhttps://wiki.math.wisc.edu/index.php?title=NTS_Fall_2011/Abstracts&diff=2970NTS Fall 2011/Abstracts2011-10-31T16:02:35Z<p>Klagsbru: </p>
<hr />
<div>== September 8 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Alexander Fish''' (Madison)<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Solvability of Diophantine equations within dynamically defined subsets of N<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: Given a dynamical system, i.e. a compact metric space ''X'', a homeomorphism ''T'' (or just a continuous map) and a Borel probability measure on ''X'' which is preserved under the action of ''T'', the dynamically defined subset associated to a point ''x'' in ''X'' and an open set ''U'' in ''X'' is {''n'' | ''T<sup>&thinsp;n</sup>''(''x'') is in ''U''} which we call the set of return times of ''x'' in ''U''. We study combinatorial properties of sets of return times for certain types of dynamical systems for generic points ''x'' in ''X''. Among examples of such sets are normal sets which correspond to the system ''X''&nbsp;=&nbsp;[0,1], ''T''(''x'')&nbsp;=&nbsp;2''x''&nbsp;mod&nbsp;1, Lebesgue measure, ''U''&nbsp;=&nbsp;[0,&nbsp;1/2].<br />
We give a complete classification of linear Diophantine systems solvable within every normal set. The methods combine the probabilistic method together with the use of van der Corput's lemma. At the end of the talk we will discuss open problems<br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== September 15 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Chung Pang Mok''' (McMaster)<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Galois representation associated to cusp forms on GL<sub>2</sub> over CM fields<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: We generalize the work of Harris–Soudry–Taylor, and constructs<br />
the compatible system of 2-dimensional ''p''-adic Galois representations<br />
associated to a cuspidal automorphic representation of cohomological type<br />
on GL<sub>2</sub> over a CM field, whose central character satisfies an invariance<br />
condition. A local-global compatibility statement, up to<br />
semi-simplification, can also be proved in this setting. This work relies<br />
crucially on Arthur's results on lifting from the group GSp<sub>4</sub> to GL<sub>4</sub>.<br />
<br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== September 22 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Yifeng Liu''' (Columbia)<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Arithmetic inner product formula<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: I will introduce an arithmetic version of the classical Rallis' inner product<br />
for unitary groups, which generalizes the previous work by Kudla, Kudla–Rapoport–Yang and<br />
Bruinier–Yang. The arithmetic inner product formula, which is still a conjecture for<br />
higher rank, relates the canonical height of special cycles on certain Shimura varieties<br />
and the central derivatives of ''L''-functions.<br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== September 29 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Nigel Boston''' (Madison)<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Non-abelian Cohen-Lenstra heuristics.<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: In 1983, Cohen and Lenstra observed that the frequency with which a given abelian ''p''-group ''A'' (''p'' odd) arises as the ''p''-class group of an imaginary quadratic field ''K'' is apparently proportional to 1/|Aut(''A'')|. The Galois group of the maximal<br />
unramified ''p''-extension of ''K'' has abelianization ''A'' and one might then ask how frequently a given ''p''-group ''G'' arises. We<br />
develop a theory wherein this frequency is inversely proportional to the size of its automorphism group in a new category<br />
and then test this against computations. If time permits, I shall describe progress on the real quadratic case. This is<br />
joint work with Michael Bush and Farshid Hajir.<br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== October 6 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Zhiwei Yun''' (MIT)<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Exceptional Lie groups as motivic Galois groups<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: More than two decades ago, Serre asked the following<br />
question: can exceptional Lie groups be realized as the motivic Galois<br />
group of some motive over a number field? The question has been open<br />
for exceptional groups other than ''G''<sub>2</sub>. In this talk, I will show how<br />
to use geometric Langlands theory to give a uniform construction of<br />
motives with motivic Galois groups ''E''<sub>7</sub>, ''E''<sub>8</sub> and ''G''<sub>2</sub>, hence giving an<br />
affirmative answer to Serre's question in these cases.<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== October 13 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Melanie Matchett Wood''' (Madison)<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: The probability that a curve over a finite field is smooth<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
<br />
Abstract: Given a fixed variety over a finite field, we ask what<br />
proportion of hypersurfaces (effective divisors) are smooth. Poonen's<br />
work on Bertini theorems over finite fields answers this question when<br />
one considers effective divisors linearly equivalent to a multiple of<br />
a fixed ample divisor, which corresponds to choosing an ample ray<br />
through the origin in the Picard group of the variety. In this case<br />
the probability of smoothness is predicted by a simple heuristic<br />
assuming smoothness is independent at different points in the ambient<br />
space. In joint work with Erman, we consider this question for<br />
effective divisors along nef rays in certain surfaces. Here the<br />
simple heuristic of independence fails, but the answer can still be<br />
determined and follows from a richer heuristic that predicts at<br />
which points smoothness is independent and at which<br />
points it is dependent.<br />
<br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== October 20 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Jie Ling''' (Madison)<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Arithmetic intersection on Toric schemes and resultants<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: Let ''K'' be a number field, ''O<sub>K</sub>'' its ring of integers. Consider ''n''&nbsp;+&nbsp;1 Laurent polynomials ''f<sub>i</sub>'' in ''n'' variables with ''O<sub>K</sub>'' coefficients. We assume that they have support in given polytopes &Delta;<sub>''i''</sub>. On one hand, we can associate a toric scheme ''X'' over ''O<sub>K</sub>'' to these polytopes, and consider the arithmetic intersection number of (''f''<sub>0</sub>,&nbsp;...,''f<sub>n</sub>'')<sub>''X''</sub> in ''X''. On the other hand, we have the mixed resultant Res(''f''<sub>0</sub>,&nbsp;...,''f<sub>n</sub>''). When the associated scheme is projective and smooth at the generic fiber and we assume ''f''<sub>0</sub>,&nbsp;...,''f<sub>n</sub>'' intersect properly, the arithmetic intersection number is given by the norm of the mixed resultant. This could be thought of as an arithmetic analog of Beinstein's theorem.<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== October 27 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Danny Neftin''' (Michigan)<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Arithmetic field relations and crossed product division algebras<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: A finite group ''G'' is called ''K''-admissible if there exists a Galois ''G''-extension ''L''/''K'' such that ''L'' is a maximal subfield of a division algebra with center ''K'' (i.e. if there exists a ''G''-crossed product ''K''-division algebra). Motivated by the works of Neukirch, Uchida, and Iwasawa, Sonn defined an arithmetic equivalence of number fields by admissibility and posed (1985) the following open problem: Do two number fields with the same admissible groups necessarily have the same degree over Q?<br />
Up until recently only positive results were known. We shall discuss equivalence by admissibility and construct two number fields that have the same odd order admissible groups, providing evidence to a negative answer to Sonn's problem.<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== November 3 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Zev Klagsburn''' (Madison)<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Selmer Ranks of Quadratic Twists of Elliptic Curves<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: Given an elliptic curve <it>E</it> defined over a number field <it>K</it>, we can ask what proportion of quadratic twists of <it>E</it> have 2-Selmer rank <it>r</it> for any non-negative integer <it>r</it>. The Delaunay heuristics combined with work of Dokchitser and Dokchitser suggested a conjecture for this distribution that was verified by work of Heath-Brown, Swinnerton-Dyer, and Kane for elliptic curves over the rationals $$ with "E(\mathbb{Q})[2] = Z/2 x Z/2". We present new results for elliptic curves with "E(K)[2] = 0" and with "E(K)[2] = Z/2". I will present joint work of Mazur, Rubin, and myself supporting the conjecture for curves with "E(K)[2] = 0" Additionally, I will present some new results of my own for curves with "E(K)[2] =Z/2", including some surprising results that conflict with the conjecture.<br />
<br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== November 10 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Luanlei Zhao''' (Madison)<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: tba<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: tba<br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== November 17 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Robert Harron''' (Madison)<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: tba<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: tba<br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== December 1 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Andrei Calderaru''' (Madison)<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: tba<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: tba<br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== December 8 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Xinwen Zhu''' (Harvard)<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: tba<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: tba<br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== December 15 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Shamgar Gurevich''' (Madison)<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Quadratic reciprocity and the sign of the Gauss sum via the finite Weil representation<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: tba<br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== Organizer contact information ==<br />
<br />
[http://www.math.wisc.edu/~shamgar/ Shamgar Gurevich]<br />
<br />
[http://www.math.wisc.edu/~rharron/ Robert Harron]<br />
<br />
[http://www.math.wisc.edu/~klagsbru/ Zev Klagsbrun]<br />
<br />
[http://www.math.wisc.edu/~mmwood/ Melanie Matchett Wood]<br />
<br><br />
<br />
<br />
<br />
----<br />
Return to the [[NTS|Number Theory Seminar Page]]<br />
<br />
Return to the [[Algebra|Algebra Group Page]]</div>Klagsbruhttps://wiki.math.wisc.edu/index.php?title=NTS_Fall_2011/Abstracts&diff=2969NTS Fall 2011/Abstracts2011-10-31T15:59:24Z<p>Klagsbru: /* November 3 */</p>
<hr />
<div>== September 8 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Alexander Fish''' (Madison)<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Solvability of Diophantine equations within dynamically defined subsets of N<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: Given a dynamical system, i.e. a compact metric space ''X'', a homeomorphism ''T'' (or just a continuous map) and a Borel probability measure on ''X'' which is preserved under the action of ''T'', the dynamically defined subset associated to a point ''x'' in ''X'' and an open set ''U'' in ''X'' is {''n'' | ''T<sup>&thinsp;n</sup>''(''x'') is in ''U''} which we call the set of return times of ''x'' in ''U''. We study combinatorial properties of sets of return times for certain types of dynamical systems for generic points ''x'' in ''X''. Among examples of such sets are normal sets which correspond to the system ''X''&nbsp;=&nbsp;[0,1], ''T''(''x'')&nbsp;=&nbsp;2''x''&nbsp;mod&nbsp;1, Lebesgue measure, ''U''&nbsp;=&nbsp;[0,&nbsp;1/2].<br />
We give a complete classification of linear Diophantine systems solvable within every normal set. The methods combine the probabilistic method together with the use of van der Corput's lemma. At the end of the talk we will discuss open problems<br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== September 15 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Chung Pang Mok''' (McMaster)<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Galois representation associated to cusp forms on GL<sub>2</sub> over CM fields<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: We generalize the work of Harris–Soudry–Taylor, and constructs<br />
the compatible system of 2-dimensional ''p''-adic Galois representations<br />
associated to a cuspidal automorphic representation of cohomological type<br />
on GL<sub>2</sub> over a CM field, whose central character satisfies an invariance<br />
condition. A local-global compatibility statement, up to<br />
semi-simplification, can also be proved in this setting. This work relies<br />
crucially on Arthur's results on lifting from the group GSp<sub>4</sub> to GL<sub>4</sub>.<br />
<br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== September 22 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Yifeng Liu''' (Columbia)<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Arithmetic inner product formula<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: I will introduce an arithmetic version of the classical Rallis' inner product<br />
for unitary groups, which generalizes the previous work by Kudla, Kudla–Rapoport–Yang and<br />
Bruinier–Yang. The arithmetic inner product formula, which is still a conjecture for<br />
higher rank, relates the canonical height of special cycles on certain Shimura varieties<br />
and the central derivatives of ''L''-functions.<br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== September 29 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Nigel Boston''' (Madison)<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Non-abelian Cohen-Lenstra heuristics.<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: In 1983, Cohen and Lenstra observed that the frequency with which a given abelian ''p''-group ''A'' (''p'' odd) arises as the ''p''-class group of an imaginary quadratic field ''K'' is apparently proportional to 1/|Aut(''A'')|. The Galois group of the maximal<br />
unramified ''p''-extension of ''K'' has abelianization ''A'' and one might then ask how frequently a given ''p''-group ''G'' arises. We<br />
develop a theory wherein this frequency is inversely proportional to the size of its automorphism group in a new category<br />
and then test this against computations. If time permits, I shall describe progress on the real quadratic case. This is<br />
joint work with Michael Bush and Farshid Hajir.<br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== October 6 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Zhiwei Yun''' (MIT)<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Exceptional Lie groups as motivic Galois groups<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: More than two decades ago, Serre asked the following<br />
question: can exceptional Lie groups be realized as the motivic Galois<br />
group of some motive over a number field? The question has been open<br />
for exceptional groups other than ''G''<sub>2</sub>. In this talk, I will show how<br />
to use geometric Langlands theory to give a uniform construction of<br />
motives with motivic Galois groups ''E''<sub>7</sub>, ''E''<sub>8</sub> and ''G''<sub>2</sub>, hence giving an<br />
affirmative answer to Serre's question in these cases.<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== October 13 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Melanie Matchett Wood''' (Madison)<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: The probability that a curve over a finite field is smooth<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
<br />
Abstract: Given a fixed variety over a finite field, we ask what<br />
proportion of hypersurfaces (effective divisors) are smooth. Poonen's<br />
work on Bertini theorems over finite fields answers this question when<br />
one considers effective divisors linearly equivalent to a multiple of<br />
a fixed ample divisor, which corresponds to choosing an ample ray<br />
through the origin in the Picard group of the variety. In this case<br />
the probability of smoothness is predicted by a simple heuristic<br />
assuming smoothness is independent at different points in the ambient<br />
space. In joint work with Erman, we consider this question for<br />
effective divisors along nef rays in certain surfaces. Here the<br />
simple heuristic of independence fails, but the answer can still be<br />
determined and follows from a richer heuristic that predicts at<br />
which points smoothness is independent and at which<br />
points it is dependent.<br />
<br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== October 20 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Jie Ling''' (Madison)<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Arithmetic intersection on Toric schemes and resultants<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: Let ''K'' be a number field, ''O<sub>K</sub>'' its ring of integers. Consider ''n''&nbsp;+&nbsp;1 Laurent polynomials ''f<sub>i</sub>'' in ''n'' variables with ''O<sub>K</sub>'' coefficients. We assume that they have support in given polytopes &Delta;<sub>''i''</sub>. On one hand, we can associate a toric scheme ''X'' over ''O<sub>K</sub>'' to these polytopes, and consider the arithmetic intersection number of (''f''<sub>0</sub>,&nbsp;...,''f<sub>n</sub>'')<sub>''X''</sub> in ''X''. On the other hand, we have the mixed resultant Res(''f''<sub>0</sub>,&nbsp;...,''f<sub>n</sub>''). When the associated scheme is projective and smooth at the generic fiber and we assume ''f''<sub>0</sub>,&nbsp;...,''f<sub>n</sub>'' intersect properly, the arithmetic intersection number is given by the norm of the mixed resultant. This could be thought of as an arithmetic analog of Beinstein's theorem.<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== October 27 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Danny Neftin''' (Michigan)<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Arithmetic field relations and crossed product division algebras<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: A finite group ''G'' is called ''K''-admissible if there exists a Galois ''G''-extension ''L''/''K'' such that ''L'' is a maximal subfield of a division algebra with center ''K'' (i.e. if there exists a ''G''-crossed product ''K''-division algebra). Motivated by the works of Neukirch, Uchida, and Iwasawa, Sonn defined an arithmetic equivalence of number fields by admissibility and posed (1985) the following open problem: Do two number fields with the same admissible groups necessarily have the same degree over Q?<br />
Up until recently only positive results were known. We shall discuss equivalence by admissibility and construct two number fields that have the same odd order admissible groups, providing evidence to a negative answer to Sonn's problem.<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== November 3 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Zev Klagsburn''' (Madison)<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Selmer Ranks of Quadratic Twists of Elliptic Curves<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: Given an elliptic curve $E$ defined over a number field $K$, we can ask what proportion of quadratic twists of $E$ have 2-Selmer rank $r$ for any non-negative integer $r$. The Delaunay heuristics combined with work of Dokchitser and Dokchitser suggested a conjecture for this distribution that was verified by work of Heath-Brown, Swinnerton-Dyer, and Kane for elliptic curves over $\mathbb{Q}$ with $E(\mathbb{Q})[2] \simeq \mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z}$. We present new results for elliptic curves with $E(K)[2] = 0$ and with $E(K)[2] \simeq \mathbb{Z}/2\mathbb{Z}$. I will present joint work of Mazur, Rubin, and myself supporting the conjecture for curves with $E(K)[2] = 0$. Additionally, I will present some new results of my own for curves with $E(K)[2] \simeq \mathbb{Z}/2\mathbb{Z}$, including some surprising results that conflict with the conjecture.<br />
<br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== November 10 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Luanlei Zhao''' (Madison)<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: tba<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: tba<br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== November 17 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Robert Harron''' (Madison)<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: tba<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: tba<br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== December 1 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Andrei Calderaru''' (Madison)<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: tba<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: tba<br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== December 8 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Xinwen Zhu''' (Harvard)<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: tba<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: tba<br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== December 15 ==<br />
<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Shamgar Gurevich''' (Madison)<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Quadratic reciprocity and the sign of the Gauss sum via the finite Weil representation<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: tba<br />
<br />
|} <br />
</center><br />
<br />
<br><br />
<br />
== Organizer contact information ==<br />
<br />
[http://www.math.wisc.edu/~shamgar/ Shamgar Gurevich]<br />
<br />
[http://www.math.wisc.edu/~rharron/ Robert Harron]<br />
<br />
[http://www.math.wisc.edu/~klagsbru/ Zev Klagsbrun]<br />
<br />
[http://www.math.wisc.edu/~mmwood/ Melanie Matchett Wood]<br />
<br><br />
<br />
<br />
<br />
----<br />
Return to the [[NTS|Number Theory Seminar Page]]<br />
<br />
Return to the [[Algebra|Algebra Group Page]]</div>Klagsbruhttps://wiki.math.wisc.edu/index.php?title=NTSGrad_Fall_2011&diff=2659NTSGrad Fall 20112011-09-19T16:04:54Z<p>Klagsbru: /* Fall 2011 Semester */</p>
<hr />
<div>= Number Theory – Representation Theory Graduate Student Seminar, University of Wisconsin–Madison =<br />
<br />
<br />
*'''When:''' Tuesdays at 2:30pm<br />
*'''Where:''' Van Vleck Hall B203<br />
<br />
The purpose of this seminar is to have a talk on each Tuesday by a graduate student to<br />
help orient ourselves for the [[NTS|Number Theory Seminar]] talk on the following Thursday.<br />
These talks should be aimed at beginning graduate students, and should try to <br />
explain some of the background, terminology, and ideas for the Thursday talk. <br />
<br />
== Fall 2011 Semester ==<br />
<br />
<center><br />
<br />
{| style="color:black; font-size:120%" border="0" cellpadding="14" cellspacing="5"<br />
|-<br />
| bgcolor="#D0D0D0" width="300" align="center"|'''Date'''<br />
| bgcolor="#F0A0A0" width="300" align="center"|'''Speaker'''<br />
| bgcolor="#BCD2EE" width="300" align="center"|'''Title (click to see abstract)'''<br />
|-<br />
| bgcolor="#E0E0E0"| September 6 (Tue.)<br />
| bgcolor="#F0B0B0"| Derek Garton<br />
| bgcolor="#BCE2FE"|[[NTS/Abstracts | <font color="black"><em>Solvability of Diophantine equations in dynamically defined sets</em></font>]]<br />
|-<br />
| bgcolor="#E0E0E0"| September 13 (Tue.)<br />
| bgcolor="#F0B0B0"| Andrew Bridy<br />
| bgcolor="#BCE2FE"|[[NTS/Abstracts | <font color="black"><em>Galois representations attached to modular and automorphic forms</em></font>]]<br />
|-<br />
| bgcolor="#E0E0E0"| September 27 (Tue.)<br />
| bgcolor="#F0B0B0"| Jonathan Blackhurst<br />
| bgcolor="#BCE2FE"|[[NTS/Abstracts | <font color="black"><em>TBA</em></font>]]<br />
|-<br />
| bgcolor="#E0E0E0"| October 4 (Tue.)<br />
| bgcolor="#F0B0B0"| Silas Johnson<br />
| bgcolor="#BCE2FE"|[[NTS/Abstracts | <font color="black"><em>TBA</em></font>]]<br />
|-<br />
| bgcolor="#E0E0E0"| October 11 (Tue.)<br />
| bgcolor="#F0B0B0"| Lalit Jain <br />
| bgcolor="#BCE2FE"|[[NTS/Abstracts | <font color="black"><em>TBA</em></font>]]<br />
|-<br />
| bgcolor="#E0E0E0"| October 18 (Tue.)<br />
| bgcolor="#F0B0B0"| Yueke Hu<br />
| bgcolor="#BCE2FE"|[[NTS/Abstracts | <font color="black"><em>TBA</em></font>]]<br />
|-<br />
| bgcolor="#E0E0E0"| October 25 (Tue.)<br />
| bgcolor="#F0B0B0"| Evan Dummit<br />
| bgcolor="#BCE2FE"|[[NTS/Abstracts | <font color="black"><em>TBA</em></font>]]<br />
|-<br />
| bgcolor="#E0E0E0"| November 1 (Tue.)<br />
| bgcolor="#F0B0B0"| Rachel Davis<br />
| bgcolor="#BCE2FE"|[[NTS/Abstracts | <font color="black"><em>TBA</em></font>]]<br />
|-<br />
| bgcolor="#E0E0E0"| November 8 (Tue.)<br />
| bgcolor="#F0B0B0"| Christelle Vincent<br />
| bgcolor="#BCE2FE"|[[NTS/Abstracts | <font color="black"><em>TBA</em></font>]]<br />
|-<br />
| bgcolor="#E0E0E0"| November 15 (Tue.)<br />
| bgcolor="#F0B0B0"| Daniel Ross<br />
| bgcolor="#BCE2FE"|[[NTS/Abstracts | <font color="black"><em>TBA</em></font>]]<br />
|-<br />
| bgcolor="#E0E0E0"| November 22 (Tue.)<br />
| bgcolor="#F0B0B0"| No seminar <br> (Thanksgiving)<br />
| bgcolor="#BCE2FE"|[[NTS/Abstracts | <font color="black"><em>Thanksgiving</em></font>]]<br />
|-<br />
| bgcolor="#E0E0E0"| November 29 (Tue.)<br />
| bgcolor="#F0B0B0"| Peyman Morteza &amp; Peng Yu<br />
| bgcolor="#BCE2FE"|[[NTS/Abstracts | <font color="black"><em>TBA</em></font>]]<br />
|-<br />
| bgcolor="#E0E0E0"| December 6 (Tue.)<br />
| bgcolor="#F0B0B0"| Marci Hablicsek<br />
| bgcolor="#BCE2FE"|[[NTS/Abstracts | <font color="black"><em>TBA</em></font>]]<br />
|-<br />
| bgcolor="#E0E0E0"| December 13 (Tue.)<br />
| bgcolor="#F0B0B0"| Daniel Ross<br />
| bgcolor="#BCE2FE"|[[NTS/Abstracts | <font color="black"><em>TBA</em></font>]]<br />
|}<br />
</center><br />
<br />
== Organizers ==<br />
[http://www.math.wisc.edu/~rharron/ Robert Harron]<br />
<br />
Zev Klagsbrun<br />
<br />
<br />
----<br />
The Spring 2011 NTS Grad page can be found [[NTSGrad Spring 2011|here]].<br />
----<br />
Return to the [[NTS|Number Theory Seminar Page]]<br />
<br />
Return to the [[Algebra|Algebra Group Page]]</div>Klagsbruhttps://wiki.math.wisc.edu/index.php?title=NTSGrad_Fall_2011&diff=2587NTSGrad Fall 20112011-09-12T19:47:53Z<p>Klagsbru: </p>
<hr />
<div>= Number Theory – Representation Theory Graduate Student Seminar, University of Wisconsin–Madison =<br />
<br />
<br />
*'''When:''' Tuesdays at 2:30pm<br />
*'''Where:''' Van Vleck Hall B203<br />
<br />
The purpose of this seminar is to have a talk on each Tuesday by a graduate student to<br />
help orient ourselves for the [[NTS|Number Theory Seminar]] talk on the following Thursday.<br />
These talks should be aimed at beginning graduate students, and should try to <br />
explain some of the background, terminology, and ideas for the Thursday talk. <br />
<br />
== Fall 2011 Semester ==<br />
<br />
<center><br />
<br />
{| style="color:black; font-size:120%" border="0" cellpadding="14" cellspacing="5"<br />
|-<br />
| bgcolor="#D0D0D0" width="300" align="center"|'''Date'''<br />
| bgcolor="#F0A0A0" width="300" align="center"|'''Speaker'''<br />
| bgcolor="#BCD2EE" width="300" align="center"|'''Title (click to see abstract)'''<br />
|-<br />
| bgcolor="#E0E0E0"| September 6 (Tue.)<br />
| bgcolor="#F0B0B0"| Derek Garton<br />
| bgcolor="#BCE2FE"|[[NTS/Abstracts | <font color="black"><em>TBA</em></font>]]<br />
|-<br />
| bgcolor="#E0E0E0"| September 13 (Tue.)<br />
| bgcolor="#F0B0B0"| Andrew Bridy<br />
| bgcolor="#BCE2FE"|[[NTS/Abstracts | <font color="black"><em>Galois Representations Attached to Modular and Automorphic Forms</em></font>]]<br />
|-<br />
| bgcolor="#E0E0E0"| September 20 (Tue.)<br />
| bgcolor="#F0B0B0"| Who?<br />
| bgcolor="#BCE2FE"|[[NTS/Abstracts | <font color="black"><em>TBA</em></font>]]<br />
|-<br />
| bgcolor="#E0E0E0"| September 27 (Tue.)<br />
| bgcolor="#F0B0B0"| Jonathan Blackhurst<br />
| bgcolor="#BCE2FE"|[[NTS/Abstracts | <font color="black"><em>TBA</em></font>]]<br />
|-<br />
| bgcolor="#E0E0E0"| October 4 (Tue.)<br />
| bgcolor="#F0B0B0"| Silas Johnson<br />
| bgcolor="#BCE2FE"|[[NTS/Abstracts | <font color="black"><em>TBA</em></font>]]<br />
|-<br />
| bgcolor="#E0E0E0"| October 11 (Tue.)<br />
| bgcolor="#F0B0B0"| Lalit Jain <br />
| bgcolor="#BCE2FE"|[[NTS/Abstracts | <font color="black"><em>TBA</em></font>]]<br />
|-<br />
| bgcolor="#E0E0E0"| October 18 (Tue.)<br />
| bgcolor="#F0B0B0"| Yueke Hu<br />
| bgcolor="#BCE2FE"|[[NTS/Abstracts | <font color="black"><em>TBA</em></font>]]<br />
|-<br />
| bgcolor="#E0E0E0"| October 25 (Tue.)<br />
| bgcolor="#F0B0B0"| Evan Dummit<br />
| bgcolor="#BCE2FE"|[[NTS/Abstracts | <font color="black"><em>TBA</em></font>]]<br />
|-<br />
| bgcolor="#E0E0E0"| November 1 (Tue.)<br />
| bgcolor="#F0B0B0"| Rachel Davis<br />
| bgcolor="#BCE2FE"|[[NTS/Abstracts | <font color="black"><em>TBA</em></font>]]<br />
|-<br />
| bgcolor="#E0E0E0"| November 8 (Tue.)<br />
| bgcolor="#F0B0B0"| Christelle Vincent<br />
| bgcolor="#BCE2FE"|[[NTS/Abstracts | <font color="black"><em>TBA</em></font>]]<br />
|-<br />
| bgcolor="#E0E0E0"| November 15 (Tue.)<br />
| bgcolor="#F0B0B0"| Daniel Ross<br />
| bgcolor="#BCE2FE"|[[NTS/Abstracts | <font color="black"><em>TBA</em></font>]]<br />
|-<br />
| bgcolor="#E0E0E0"| November 22 (Tue.)<br />
| bgcolor="#F0B0B0"| No seminar <br> (Thanksgiving)<br />
| bgcolor="#BCE2FE"|[[NTS/Abstracts | <font color="black"><em>Thanksgiving</em></font>]]<br />
|-<br />
| bgcolor="#E0E0E0"| November 29 (Tue.)<br />
| bgcolor="#F0B0B0"| Peyman Morteza &amp; Peng Yu<br />
| bgcolor="#BCE2FE"|[[NTS/Abstracts | <font color="black"><em>TBA</em></font>]]<br />
|-<br />
| bgcolor="#E0E0E0"| December 6 (Tue.)<br />
| bgcolor="#F0B0B0"| Marci Hablicsek<br />
| bgcolor="#BCE2FE"|[[NTS/Abstracts | <font color="black"><em>TBA</em></font>]]<br />
|-<br />
| bgcolor="#E0E0E0"| December 13 (Tue.)<br />
| bgcolor="#F0B0B0"| Daniel Ross<br />
| bgcolor="#BCE2FE"|[[NTS/Abstracts | <font color="black"><em>TBA</em></font>]]<br />
|}<br />
</center><br />
<br />
== Organizers ==<br />
[http://www.math.wisc.edu/~rharron/ Robert Harron]<br />
<br />
Zev Klagsbrun<br />
<br />
<br />
----<br />
The Spring 2011 NTS Grad page can be found [[NTSGrad Spring 2011|here]].<br />
----<br />
Return to the [[NTS|Number Theory Seminar Page]]<br />
<br />
Return to the [[Algebra|Algebra Group Page]]</div>Klagsbruhttps://wiki.math.wisc.edu/index.php?title=NTS_Fall_2011&diff=2379NTS Fall 20112011-08-29T04:26:32Z<p>Klagsbru: </p>
<hr />
<div>= Number Theory – Representation Theory Seminar, University of Wisconsin–Madison =<br />
<br />
<br />
*'''When:''' Thursdays at 2:30pm.<br />
*'''Where:''' Van Vleck Hall B203<br />
*Please join the [https://mailhost.math.wisc.edu/mailman/listinfo/nts NT/RT mailing list:]. <br />
<br />
<br />
<br />
== Fall 2011 Semester ==<br />
<br />
<center><br />
<br />
{| style="color:black; font-size:120%" border="0" cellpadding="14" cellspacing="5"<br />
|-<br />
| bgcolor="#D0D0D0" width="300" align="center"|'''Date'''<br />
| bgcolor="#F0A0A0" width="300" align="center"|'''Speaker'''<br />
| bgcolor="#BCD2EE" width="300" align="center"|'''Title (click to see abstract)'''<br />
|-<br />
| bgcolor="#E0E0E0"| September 8 (Thurs.)<br />
| bgcolor="#F0B0B0"| [http://www.math.wisc.edu/~afish/ Alexander Fish] <br> (Madison)<br />
| bgcolor="#BCE2FE"|[[NTS Fall 2011/Abstracts#September 8 | <font color="black"><em>Solvability of Diophantine equations within dynamically defined subsets of N</em></font>]]<br />
|-<br />
| bgcolor="#E0E0E0"| September 15 (Thurs.)<br />
| bgcolor="#F0B0B0"| [http://www.math.cuhk.edu.hk/~cpmok/ Chung Pang Mok] <br> (McMaster)<br />
| bgcolor="#BCE2FE"|[[NTS Fall 2011/Abstracts#September 15 | <font color="black"><em> Galois representation associated to cusp forms on GL(2) over CM fields</em></font>]]<br />
|-<br />
| bgcolor="#E0E0E0"| September 22 (Thurs.)<br />
| bgcolor="#F0B0B0"| [http://math.columbia.edu/~liuyf/ Yifeng Liu] <br> (Columbia U.)<br />
| bgcolor="#BCE2FE"|[[NTS Fall 2011/Abstracts#September 22 | <font color="black"><em>Problem and Solution</em></font>]]<br />
|-<br />
| bgcolor="#E0E0E0"| September 29 (Thurs.)<br />
| bgcolor="#F0B0B0"| [http://www.math.wisc.edu/~boston/ Nigel Boston] <br> (Madison)<br />
| bgcolor="#BCE2FE"|[[NTS Fall 2011/Abstracts#September 29 | <font color="black"><em>Problem and Solution</em></font>]]<br />
|-<br />
| bgcolor="#E0E0E0"| October 6 (Thurs.)<br />
| bgcolor="#F0B0B0"| [http://math.mit.edu/~zyun/ Zhiwei Yun] <br> (MIT)<br />
| bgcolor="#BCE2FE"|[[NTS Fall 2011/Abstracts#October 6 | <font color="black"><em>Problem and Solution</em></font>]]<br />
|-<br />
| bgcolor="#E0E0E0"| October 13 (Thurs.)<br />
| bgcolor="#F0B0B0"| [http://math.stanford.edu/~mwood/ Melanie Matchett Wood] <br> (Madison)<br />
| bgcolor="#BCE2FE"|[[NTS Fall 2011/Abstracts | <font color="black"><em>The probability that a curve over a finite field is smooth</em></font>]]<br />
|-<br />
| bgcolor="#E0E0E0"| October 20 (Thurs.)<br />
| bgcolor="#F0B0B0"| [http://www.math.wisc.edu/~ling/ Jie Ling] <br> (UW-Madison)<br />
| bgcolor="#BCE2FE"|[[NTS Fall 2011/Abstracts | <font color="black"><em>Problem and Solution</em></font>]]<br />
|-<br />
| bgcolor="#E0E0E0"| October 27 (Thurs.)<br />
| bgcolor="#F0B0B0"| Who <br> (from where)<br />
| bgcolor="#BCE2FE"|[[NTS Fall 2011/Abstracts | <font color="black"><em>Problem and Solution</em></font>]]<br />
|-<br />
| bgcolor="#E0E0E0"| November 3 (Thurs.)<br />
| bgcolor="#F0B0B0"| Zev Klagsbrun <br> (Madison)<br />
| bgcolor="#BCE2FE"|[[NTS Fall 2011/Abstracts#November 3 | <font color="black"><em>Problem and Solution</em></font>]]<br />
|-<br />
| bgcolor="#E0E0E0"| November 10 (Thurs.)<br />
| bgcolor="#F0B0B0"| [http://www.math.wisc.edu/~apache/gallery/grad.html Luanlei Zhao] <br> (UW-Madison)<br />
| bgcolor="#BCE2FE"|[[NTS Fall 2011/Abstracts | <font color="black"><em>Problem and Solution</em></font>]]<br />
|-<br />
| bgcolor="#E0E0E0"| November 17 (Thurs.)<br />
| bgcolor="#F0B0B0"| [http://math.bu.edu/people/rharron/ Robert Harron] <br> (Madison)<br />
| bgcolor="#BCE2FE"|[[NTS Fall 2011/Abstracts#November 17 | <font color="black"><em>Problem and Solution</em></font>]]<br />
|-<br />
| bgcolor="#E0E0E0"| November 24 (Thurs.)<br />
| bgcolor="#F0B0B0"| No seminar <br> (Thanksgiving)<br />
| bgcolor="#BCE2FE"| <font color="black"><em>Thanksgiving</em></font><br />
|-<br />
| bgcolor="#E0E0E0"| December 1 (Thurs.)<br />
| bgcolor="#F0B0B0"| [http://www.math.wisc.edu/~andreic/ Andrei Calderaru] <br> (Madison)<br />
| bgcolor="#BCE2FE"|[[NTS Fall 2011/Abstracts | <font color="black"><em>Problem and Solution</em></font>]]<br />
|- <br />
| bgcolor="#E0E0E0"| December 8 (Thurs.)<br />
| bgcolor="#F0B0B0"| [http://www.math.harvard.edu/~xinwenz/ Xinwen Zhu] <br> (Harvard)<br />
| bgcolor="#BCE2FE"|[[NTS Fall 2011/Abstracts#December 8 | <font color="black"><em>Problem and Solution</em></font>]]<br />
|- <br />
| bgcolor="#E0E0E0"| December 15 (Thurs.)<br />
| bgcolor="#F0B0B0"| [http://www.math.wisc.edu/~shamgar/ Shamgar Gurevich] <br> (Madison)<br />
| bgcolor="#BCE2FE"|[[NTS Fall 2011/Abstracts | <font color="black"><em> Quadratic reciprocity and the sign of the Gauss sum via the finite Weil representation </em></font>]]<br />
|}<br />
<br />
</center><br />
<br />
<br><br />
<br />
== Organizer contact information ==<br />
[http://www.math.wisc.edu/~shamgar/ Shamgar Gurevich]<br />
<br />
[http://math.bu.edu/people/rharron/ Robert Harron]<br />
<br />
Zev Klagsbrun<br />
<br />
[http://math.stanford.edu/~mwood/ Melanie Matchett Wood]<br />
<br />
<br />
<br />
----<br />
Also of interest is the [[NTSGrad Fall 2011|Grad student seminar]] which meets on Tuesdays.<br><br />
Next semester's seminar page is [[NTS Spring 2012|here]].<br />
Last semester's seminar page is [[NTS spring 2011|here]].<br />
----<br />
Return to the [[Algebra|Algebra Group Page]]</div>Klagsbru