NTS Spring 2012/Abstracts: Difference between revisions
m (→March 8: minor changes) |
(→March 29: add title and abstract for David Roberts' talk) |
||
Line 259: | Line 259: | ||
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''David P. Roberts''' (U. Minnesota Morris) | | bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''David P. Roberts''' (U. Minnesota Morris) | ||
|- | |- | ||
| bgcolor="#BCD2EE" align="center" | Title: | | bgcolor="#BCD2EE" align="center" | Title: Lightly ramified number fields with Galois group ''S''.''M''<sub>12</sub>.''A'' | ||
|- | |- | ||
| bgcolor="#BCD2EE" | | | bgcolor="#BCD2EE" | | ||
Abstract: Two of the most important invariants of an irreducible polynomial ''f''(''x'') ∈ '''Z'''[''x'' ] are its | |||
Galois group ''G'' and its field discriminant ''D''. The inverse Galois problem asks | |||
one to find a polynomial ''f''(''x'') having any prescribed Galois group ''G''. Refinements | |||
of this problem ask for ''D'' to be small in various senses, for example of the form | |||
± ''p<sup>a''</sup> for the smallest possible prime ''p''. | |||
The talk will discuss this problem in general, with a focus on the technique of | |||
specializing three-point covers for solving instances of it. Then it will pursue the cases of the | |||
Mathieu group ''M''<sub>12</sub>, its automorphism group ''M''<sub>12</sub>.2, its double cover | |||
2.''M''<sub>12</sub>, and the combined extension 2.''M''<sub>12</sub>.2. Among the polynomials | |||
found is | |||
::''f''(''x'') = ''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> − 4174 ''e''<sup>7</sup> ''x''<sup>26</sup> + 11287 ''e''<sup>9</sup> ''x''<sup>24</sup> | |||
::− 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> | |||
::+ 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>, | |||
with ''e'' = 11. This polynomial has Galois group ''G'' = 2.''M''<sub>12</sub>.2 and | |||
field discriminant 11<sup>88</sup>. It makes ''M''<sub>12</sub> the | |||
first of the twenty-six sporadic simple groups Γ | |||
known to have a polynomial with Galois group | |||
''G'' involving Γ and field discriminant ''D'' | |||
the power of single prime dividing |Γ |. | |||
|} | |} |
Revision as of 02:17, 1 March 2012
February 2
Evan Dummit (Madison) |
Title: Kakeya sets over non-archimedean local rings |
Abstract: In a forthcoming paper with Marci Habliscek, we constructed a Kakeya set over the formal power series ring F_{q}[[t ]], 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. |
February 16
Tonghai Yang (Madison) |
Title: A little linear algebra on CM abelian surfaces |
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. |
February 23
Christelle Vincent (Madison) |
Title: Drinfeld modular forms |
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_{q}[T ], and about Drinfeld quasi-modular forms. |
March 1
Shamgar Gurevich (Madison) |
Title: Computing the Matched Filter in Linear Time |
Abstract: In the digital radar problem we design a function (waveform) S(t) in the Hilbert space H = C(Z/p) of complex valued functions on Z/p = {0, ..., p − 1}, the integers modulo a prime number p ≫ 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 R(t) = exp{2πiωt/p}⋅S(t+τ) + W(t), where W(t) in H is a white noise, and τ, ω in Z/p, encode the distance from, and velocity of, the object. Problem (digital radar problem) Extract τ, ω from R and S. 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^{2}⋅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. I will demonstrate additional applications to mobile communication, and global positioning system (GPS). 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). The lecture is suitable for general math/engineering audience. |
March 8
Zev Klagsbrun (Madison) |
Title: Erdős–Kac Type Theorems |
Abstract: In it's 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. The lecture is suitable for general math audience. |
March 15
Yongqiang Zhao (Madison) |
Title: tba |
Abstract: tba |
March 22
Paul Terwilliger (Madison) |
Title: tba |
Abstract: tba |
March 29
David P. Roberts (U. Minnesota Morris) |
Title: Lightly ramified number fields with Galois group S.M_{12}.A |
Abstract: Two of the most important invariants of an irreducible polynomial f(x) ∈ Z[x ] are its Galois group G and its field discriminant D. The inverse Galois problem asks one to find a polynomial f(x) having any prescribed Galois group G. Refinements of this problem ask for D to be small in various senses, for example of the form ± p^{a} for the smallest possible prime p. The talk will discuss this problem in general, with a focus on the technique of specializing three-point covers for solving instances of it. Then it will pursue the cases of the Mathieu group M_{12}, its automorphism group M_{12}.2, its double cover 2.M_{12}, and the combined extension 2.M_{12}.2. Among the polynomials found is
with e = 11. This polynomial has Galois group G = 2.M_{12}.2 and field discriminant 11^{88}. It makes M_{12} the first of the twenty-six sporadic simple groups Γ known to have a polynomial with Galois group G involving Γ and field discriminant D the power of single prime dividing |Γ |. |
April 12
Chenyan Wu (Minnesota) |
Title: tba |
Abstract: tba |
April 19
Robert Guralnick (U. Southern California) |
Title: A variant of Burnside and Galois representations which are automorphic |
Abstract: Wiles, Taylor, Harris and others used the notion of a big representation of a finite group to show that certain representations are automorphic. Jack Thorne recently observed that one could weaken this notion of bigness to get the same conclusions. He called this property adequate. An absolutely irreducible representation V of a finite group G in characteristic p is called adequate if G has no p-quotients, the dimension of V is prime to p, V has non-trivial self extensions and End(V) is generated by the linear span of the elements of order prime to p in G. If G has order prime to p, all of these conditions hold—the last condition is sometimes called Burnside's Lemma. We will discuss a recent result of Guralnick, Herzig, Taylor and Thorne showing that if p > 2 dim V + 2, then any absolutely irreducible representation is adequate. We will also discuss some examples showing that the span of the p'-elements in End(V) need not be all of End(V). |
April 26
Frank Thorne (U. South Carolina) |
Title: tba |
Abstract: tba |
May 3
Alina Cojocaru (U. Illinois at Chicago) |
Title: tba |
Abstract: tba |
May 10
Samit Dasgupta (UC Santa Cruz) |
Title: tba |
Abstract: tba |
Organizer contact information
Return to the Number Theory Seminar Page
Return to the Algebra Group Page