NTS/Abstracts: Difference between revisions
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{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20" | {| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20" | ||
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| bgcolor="# | | bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Robert Lemke Oliver''' | ||
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| bgcolor="# | | bgcolor="#BCD2EE" align="center" | ''The distribution of 2-Selmer groups of elliptic curves with two-torsion'' | ||
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| bgcolor="#BCD2EE" | | |||
Bhargava and Shankar have shown that the average size of the 2-Selmer group of an elliptic curve over Q, when curves are ordered by height, is exactly 3, and Bhargava and Ho have shown that, in the family of curves with a marked point, the average is exactly 6. In stark contrast to these results, we show that the average size in the family of elliptic curves with a two-torsion point is unbounded. This follows from an understanding of the Tamagawa ratio associated to such elliptic curves, which we prove is "normally distributed with infinite variance". This work is joint with Zev Klagsbrun. | |||
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== Sep 04 == | |||
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{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20" | |||
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| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Patrick Allen''' | |||
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| bgcolor="#BCD2EE" align="center" | ''Unramified deformation rings'' | |||
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| bgcolor="#BCD2EE" | | |||
Class field theory allows one to precisely understand ramification in abelian extensions of number fields. A consequence is that infinite pro-p abelian extensions of a number field are infinitely ramified above p. Boston conjectured a nonabelian analogue of this fact, predicting that certain universal p-adic representations that are unramified at p act via a finite quotient, and this conjecture strengthens the unramified version of the Fontaine-Mazur conjecture. We show in many cases that one can deduce Boston's conjecture from the unramified Fontaine-Mazur conjecture, which allows us to deduce (unconditionally) Boston's conjecture in many two-dimensional cases. This is joint work with F. Calegari. | |||
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== Sep 11 == | |||
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{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20" | |||
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| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Melanie Matchett Wood''' | |||
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| bgcolor="#BCD2EE" align="center" | The distribution of sandpile groups of random graphs *** | |||
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| bgcolor="#BCD2EE" | | |||
The sandpile group is an abelian group associated to a graph, given as the cokernel of the graph Laplacian. An Erdős–Rényi random graph then gives some distribution of random abelian groups. We will give an introduction to various models of random finite abelian groups arising in number theory and the connections to the distribution conjectured by Payne et. al. for sandpile groups. We will talk about the moments of random finite abelian groups, and how in practice these are often more accessible than the distributions themselves, but frustratingly are not a priori guaranteed to determine the distribution. In this case however, we have found the moments of the sandpile groups of random graphs, and proved they determine the measure, and have proven Payne's conjecture. | |||
*** ''This is officially a '''probability seminar''', but will occur in the usual NTS room B105 at a slightly '''earlier time''', 2:25 PM.'' | |||
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== Sep 18 == | |||
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{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20" | |||
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| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Takehiko Yasuda''' | |||
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| bgcolor="#BCD2EE" align="center" | ''Distributions of rational points and number fields, and height zeta functions'' | |||
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| bgcolor="#BCD2EE" | | |||
In this talk, I will talk about my attempt to relate Malle's conjecture on the distribution of number fields with Batyrev and Tschinkel's generalization of Manin's conjecture on the distribution of rational points on singular Fano varieties. The main tool for relating these is the height zeta function. | |||
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== Sep 25 == | |||
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{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20" | |||
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| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Ramin Takloo-Bigash''' | |||
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| bgcolor="#BCD2EE" align="center" | ''Counting orders in number fields and p-adic integrals'' | |||
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| bgcolor="#BCD2EE" | | |||
In this talk I will report on a recent work on the distribution of orders in number fields. In particular, I will sketch the proof of an asymptotic formula for the number of orders of bounded | |||
discriminant in a given quintic number field emphasizing the role played by p-adic (and motivic) integration. This is joint work with Nathan Kaplan (Yale) and Jake Marcinek (Caltech). | |||
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== Oct 02 == | |||
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{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20" | |||
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| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Pham Huu Tiep''' | |||
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| bgcolor="#BCD2EE" align="center" | ''Nilpotent Hall and abelian Hall subgroups'' | |||
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| bgcolor="#BCD2EE" | | |||
To which extent can one generalize the Sylow theorems? One possible direction is to assume the existence of a nilpotent subgroup whose order and index are coprime. We will discuss recent | |||
joint work with various collaborators that gives a criterion to detect the existence of such subgroups in any finite group. | |||
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== Oct 09 == | |||
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{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20" | |||
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| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Michael Woodbury''' | |||
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| bgcolor="#BCD2EE" align="center" | ''An Adelic Kuznetsov Trace Formula for GL(4)'' | |||
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| bgcolor="#BCD2EE" | | |||
An important tool in analytic number theory for GL(2)-type questions is Kuznetsov’s trace formula. Recently, in work of Blomer and of Goldfeld/Kontorovich, generalizations of this to GL(3) have been given which are useful for number theoretic applications. In my talk I will discuss joint work with Dorian Goldfeld in which we further generalize the said GL(3) results to GL(4). I will discuss some of the new features and complications which arise for GL(4) as well as applications to low lying zeros of L-functions and a vertical Sato-Tate theorem. | |||
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</center> | |||
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== Oct 16 == | |||
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{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20" | |||
|- | |||
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Robert Grizzard''' | |||
|- | |||
| bgcolor="#BCD2EE" align="center" | ''Small points and free abelian groups'' | |||
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| bgcolor="#BCD2EE" | | |||
Let F be an algebraic extension of the rational numbers and E an elliptic curve defined over some number field contained in F. The absolute logarithmic Weil height, respectively the Néron-Tate height, induces a norm on F* modulo torsion, respectively on E(F) modulo torsion. The groups F* and E(F) are free abelian modulo torsion if the height function does not attain arbitrarily small positive values. We prove the failure of the converse to this statement by explicitly constructing counterexamples. This is joint work with Philipp Habegger and Lukas Pottmeyer. | |||
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== Oct 23 == | |||
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{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20" | |||
|- | |||
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Tonghai Yang''' | |||
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| bgcolor="#BCD2EE" align="center" | ''A conjecture of Colmez'' | |||
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| bgcolor="#BCD2EE" | | |||
In his seminal work on the Mordell conjecture, Faltings introduces and studies the (semistable) height of an Abelian variety. When the Abelian veriety is a CM elliptic curve, its Faltings height is essentially the local derivative (at the critical point s=1) of the Dirichlet L-series associated to the imaginary quadratic field by the famous Chowla-Selberg formula. In 1990s, Colmez gave a precise conjectural formula to compute the Faltings height of a CM abelian variety of CM type (E, Φ) in terms of the log derivative at s=1 of some `Artin' L-function associated to the CM type Φ. He proved the conjecture when the CM number field when E is abelian, refining Gross and Anderson's work on periods. Around 2007, I proved the first non-abelian case of the Colmez conjecture using a totally different method--arithmetic intersection and Borcherds product. In this talk, I will talk about its generalization to a new family of CM type, which is an ongoing joint work with Bruinier, Howard, Kudla, and Rapoport. | |||
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== Oct 30 == | |||
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{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20" | |||
|- | |||
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Laura DeMarco''' | |||
|- | |||
| bgcolor="#BCD2EE" align="center" | ''Elliptic curves and complex dynamics'' | |||
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| bgcolor="#BCD2EE" | | |||
I will discuss relations between the dynamics of complex rational functions and the arithmetic of elliptic curves. My goal is to present some new work (in progress) that reproves/generalizes a 1959 result of Lang and Neron about rational points on elliptic curves over function fields. On the dynamical side, the same ideas lead to a characterization of stability for families of rational maps on P<sup>1</sup>. | |||
|} | |||
</center> | |||
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== Nov 06 == | |||
<center> | |||
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20" | |||
|- | |||
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Michael Magee''' | |||
|- | |||
| bgcolor="#BCD2EE" align="center" | ''Zero sets of Hecke polynomials on the sphere'' | |||
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| bgcolor="#BCD2EE" | | |||
The eigenspaces of the Laplacian on the two dimensional sphere consist of homogeneous polynomials and occur with increasing dimension as the eigenvalue grows. I'll explain how one can remove this high multiplicity by using arithmetic Hecke operators that arise from the Hamilton quaternions. The resulting Hecke eigenfunctions are subject to predictions arising from random function theory and quantum chaos, in particular concerning the topology of their zero sets. I'll discuss what is known in this area and how one can try to study these 'nodal lines'. | |||
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</center> | |||
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== Nov 13 == | |||
<center> | |||
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20" | |||
|- | |||
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Yiwei She''' | |||
|- | |||
| bgcolor="#BCD2EE" align="center" | ''The Shafarevich conjecture for K3 surfaces'' | |||
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| bgcolor="#BCD2EE" | | |||
Let K be a number field, S a finite set of places of K, and g a positive integer. Shafarevich made the following conjecture for higher genus curves: the set of isomorphism classes of genus g curves defined over K and with good reduction outside of S is finite. In 1983, Faltings proved this conjecture for curves and the analogous conjecture for polarized abelian surfaces. Building on the work of Faltings and Andre, we prove the stronger unpolarized Shafarevich conjecture for K3 surfaces. I will also explain the connections between the Shafarevich conjecture and the Tate conjecture. | |||
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</center> | |||
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== Nov 20 == | |||
<center> | |||
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20" | |||
|- | |||
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Simon Marshall''' | |||
|- | |||
| bgcolor="#BCD2EE" align="center" | ''Endoscopy and cohomology of unitary groups'' | |||
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| bgcolor="#BCD2EE" | | |||
We will give a rough outline of the endoscopic classification of representations of quasi-split unitary groups carried out by Mok, following Arthur and others. We will show how this can be used to prove asymptotics for the L<sup>2</sup> Betti numbers of families of locally symmetric spaces. | |||
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</center> | |||
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== Dec 04 == | |||
<center> | |||
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20" | |||
|- | |||
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Joel Specter''' | |||
|- | |||
| bgcolor="#BCD2EE" align="center" | ''Commuting Endomorphisms of the p-adic Unit Disk'' | |||
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| bgcolor="#BCD2EE" | | |||
When can a pair of endomorphisms of <math>\mathbf{Z}_p[[X]]/\mathbf{Z}_p</math> commute? Approaching this problem from the vantage point of dynamics on the p-adic unit disk, Lubin proved that whenever a non-invertible endomorphism f commutes with a non-torsion automorphism u, the pair f and u exhibit many of the same properties as endomorphisms of a formal group over <math>\mathbf{Z}_p</math>. Because of this, he posited that for such a pair of endomorphisms to exist, there in fact had to be a formal group 'somehow in the background.' In this talk, I will discuss how some of the dynamical systems of Lubin occur naturally as the restriction of the Galois action on certain Fontaine period rings. Using this observation, I will construct, in some cases, the formal groups conjectured by Lubin. | |||
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== Dec 11 == | |||
<center> | |||
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20" | |||
|- | |||
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Ila Varma''' | |||
|- | |||
| bgcolor="#BCD2EE" align="center" | ''The mean number of 3-torsion elements in ray class groups of quadratic fields'' | |||
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| bgcolor="#BCD2EE" | | |||
In 1971, Davenport and Heilbronn determined the mean number of 3-torsion elements in the | |||
class groups of quadratic fields, when ordered by discriminant. I will describe some aspects of the proof of Davenport and Heilbronn’s theorem; in particular, they prove a relationship via class field theory between the number of 3-torsion ideal classes of quadratic fields and the number of nowhere totally ramified cubic fields over <math>\mathbb{Q}</math>. This argument generalizes to give a relationship between 3-torsion elements of the ray class groups of quadratic fields and certain pairs of cubic fields satisfying explicit ramification conditions. I will illustrate how the combination of this fact with Davenport-Heilbronn’s asymptotics on the number of cubic fields of bounded discriminant allows one to compute the mean number of 3-torsion elements in ray class groups of quadratic fields. If time permits, I will discuss the analogous theorems computing the mean size of 2-torsion elements in ray class groups of cubic fields ordered by discriminant, generalizing Bhargava. | |||
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<!-- | |||
== October 3 == | |||
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{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20" | |||
|- | |||
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Andrew Bridy''' (Madison) | |||
|- | |||
| bgcolor="#BCD2EE" align="center" | Title: The Artin–Mazur zeta function of a Lattes map in positive characteristic | |||
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| bgcolor="#BCD2EE" | | |||
Abstract: The Artin–Mazur zeta function of a dynamical system is a generating function that captures information about its periodic points. In characteristic zero, the zeta function of a rational map from '''P'''<sup>1</sup> to '''P'''<sup>1</sup> is known to always be a rational function. In positive characteristic, the situation is much less clear. Lattes maps are rational maps on '''P'''<sup>1</sup> that are finite quotients of endomorphisms of elliptic curves, and they have many interesting dynamical properties related to the geometry and arithmetic of elliptic curves. I show that the zeta function of a separable Lattes map in positive characteristic is a transcendental function. | |||
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</center> | |||
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== October 10 == | |||
<center> | |||
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20" | |||
|- | |||
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Bogdan Petrenko''' (Eastern Illinois University) | |||
|- | |||
| bgcolor="#BCD2EE" align="center" | Title: Generating an algebra from the probabilistic standpoint | |||
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| bgcolor="#BCD2EE" | | |||
Abstract: Let ''A'' be a ring whose additive group is free Abelian of finite | |||
rank. The topic of this talk is the following question: what is the | |||
probability that several random elements of ''A'' generate it as a ring? After | |||
making this question precise, I will show that it has an interesting | |||
answer which can be interpreted as a local-global principle. Some | |||
applications will be discussed. This talk will be based on my joint work | |||
with Rostyslav Kravchenko (University of Chicago) and Marcin Mazur | |||
(Binghamton University). | |||
|} | |||
</center> | |||
<br> | |||
== October 17 == | |||
<center> | |||
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20" | |||
|- | |||
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Anthony Várilly-Alvarado''' (Rice) | |||
|- | |||
| bgcolor="#BCD2EE" align="center" | Title: Arithmetic of del Pezzo surfaces of degree 4 and vertical Brauer groups | |||
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| bgcolor="#BCD2EE" | | |||
Abstract: Del Pezzo surfaces X of degree 4 are smooth (complete) intersections of two quadrics in four-dimensional projective space. They are some of the simplest surfaces for which there can be cohomological obstructions to the existence of rational points, mediated by the Brauer group Br X of the surface. I will explain how to construct, for every non-trivial, non-constant element A of Br X, a rational genus-one fibration X -> P^1 such that A is "vertical" for this map. This implies, for example, that if there is a cohomological obstruction to the existence of a point on X, then there is a genus-one fibration X -> P^1 where none of the fibers are locally soluble, giving a concrete, geometric way of "seeing" a Brauer-Manin obstruction. The construction also gives a fast, practical algorithm for computing the Brauer group of X. Conjecturally, this gives a mechanical way of testing for the existence of rational points on these surfaces. This is joint work with Bianca Viray. | |||
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</center> | |||
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== October 24 == | |||
<center> | |||
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20" | |||
|- | |||
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Paul Garrett''' (Minnesota) | |||
|- | |||
| bgcolor="#BCD2EE" align="center" | Title: Spectra of pseudo-Laplacians on spaces of automorphic forms | |||
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| bgcolor="#BCD2EE" | | |||
Abstract: Faddeev–Pavlov and Lax–Phillips observed that certain | |||
restrictions of the Laplacian to parts of automorphic continuous | |||
spectrum have discrete spectrum. Colin de Verdiere used this to prove | |||
meromorphic continuation of Eisenstein series, and proposed ways to | |||
exploit this idea to construct self-adjoint operators with spectra | |||
related to zeros of ''L''-functions. We show that simple forms of this | |||
construction produce at most very sparse spectra, due to | |||
incompatibility with pair correlations for zeros. Ways around some of | |||
the obstacles are sketched. (Joint with E. Bombieri.) | |||
|} | |||
</center> | |||
<br> | |||
== October 31 == | |||
<center> | |||
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20" | |||
|- | |||
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Jerry Wang''' (Princeton) | |||
|- | |||
| bgcolor="#BCD2EE" align="center" | Title: Pencils of quadrics and the arithmetic of hyperelliptic curves | |||
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| bgcolor="#BCD2EE" | | |||
Abstract: In recent joint works with Manjul Bhargava and Benedict Gross, we showed that a positive proportion of hyperelliptic curves over '''Q''' of genus ''g'' have no points over any odd degree extension of '''Q'''. This is done by computing certain 2-Selmer averages and applying a result of Dokchitser–Dokchitser on the parity of the rank of the 2-Selmer groups in biquadratic twists. In this talk, we will see how arithmetic invariant theory and the geometric theory of pencils of quadrics are used to obtain the 2-Selmer averages. | |||
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</center> | |||
<br> | |||
== November 7 == | |||
<center> | |||
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20" | |||
|- | |||
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''who?''' (where?) | |||
|- | |||
| bgcolor="#BCD2EE" align="center" | Title: tba | |||
|- | |||
| bgcolor="#BCD2EE" | | |||
Abstract: tba | |||
|} | |||
</center> | |||
<br> | |||
== November 12 == | |||
<center> | |||
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20" | |||
|- | |||
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Evan Dummit''' (Madison) | |||
|- | |||
| bgcolor="#BCD2EE" align="center" | Title: Counting extensions of number fields of given degree, bounded (rho)-discriminant, and specified Galois closure | |||
|- | |||
| bgcolor="#BCD2EE" | | |||
Abstract: A very basic question in algebraic number theory is: how many number fields are there? A natural way to order the fields of a fixed degree n is by discriminant, and classical results of Minkowski then assure us that there are only finitely many fields with a given discriminant. We are also often interested in counting number fields, or relative extensions, with other properties, such as having a particular Galois closure. A folk conjecture sometimes attributed to Linnik states that the number of extensions of degree n and absolute discriminant less than X is on the order of X. A great deal of recent and ongoing work has been focused towards achieving upper bounds on counts of this nature (quite successfully, in degree 5 and lower), but there is comparatively little known in higher degrees, for relative extensions, or for sufficiently complicated Galois closures: the primary results are those of Schmidt and Ellenberg-Venkatesh. I will discuss these results and my thesis work, in which I generalize several of their results and introduce another counting metric, the "rho-discriminant". | |||
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</center> | |||
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== November 21 == | |||
<center> | |||
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20" | |||
|- | |||
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Michael Lipnowski''' (Duke) | |||
|- | |||
| bgcolor="#BCD2EE" align="center" | Title: tba | |||
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| bgcolor="#BCD2EE" | | |||
Abstract: tba | |||
|} | |||
</center> | |||
<br> | |||
== November 26 == | |||
<center> | |||
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20" | |||
|- | |||
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Daniel Kane''' (Stanford) | |||
|- | |||
| bgcolor="#BCD2EE" align="center" | Title: Diffuse decompositions of polynomials | |||
|- | |||
| bgcolor="#BCD2EE" | | |||
Abstract: We study some problems relating to polynomials evaluated | |||
either at random Gaussian or random Bernoulli inputs. We present some | |||
new work on a structure theorem for degree-''d'' polynomials with Gaussian | |||
inputs. In particular, if ''p'' is a given degree-''d'' polynomial, then ''p'' | |||
can be written in terms of some bounded number of other polynomials | |||
''q''<sub>1</sub>, ..., ''q''<sub>''m''</sub> so that the joint probability density function of | |||
''q''<sub>1</sub>(''G''), ..., ''q''<sub>''m''</sub>(''G'') is close to being bounded. This says essentially | |||
that any abnormalities in the distribution of ''p''(''G'') can be explained by | |||
the way in which ''p'' decomposes into the ''q''<sub>''i''</sub>. We then present some | |||
applications of this result. | |||
|} | |} | ||
</center> | </center> | ||
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== | == December 5 == | ||
<center> | |||
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20" | |||
|- | |||
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Jennifer Park''' (MIT) | |||
|- | |||
| bgcolor="#BCD2EE" align="center" | Title: tba | |||
|- | |||
| bgcolor="#BCD2EE" | | |||
Abstract: tba | |||
|} | |||
</center> | |||
<br> | |||
== December 12 == | |||
<center> | |||
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20" | |||
|- | |||
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Vivek Shende''' (Berkeley) | |||
|- | |||
| bgcolor="#BCD2EE" align="center" | Title: Equidistribution on the space of rank two vector bundles over the projective line | |||
|- | |||
| bgcolor="#BCD2EE" | | |||
Abstract: I will discuss how the algebraic geometry of hyperelliptic curves gives an approach to a function field analogue of the 'mixing conjecture' of Michel and Venkatesh. (For a rather longer abstract, see the [http://arxiv.org/abs/1307.8237 arxiv posting] of the same name as the talk). This talk presents joint work with Jacob Tsimerman. | |||
|} | |||
</center> | |||
<br> | <br> | ||
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== Organizer contact information == | |||
Sean Rostami (srostami@math.wisc.edu) | |||
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Latest revision as of 19:53, 28 November 2014
Aug 28
Robert Lemke Oliver |
The distribution of 2-Selmer groups of elliptic curves with two-torsion |
Bhargava and Shankar have shown that the average size of the 2-Selmer group of an elliptic curve over Q, when curves are ordered by height, is exactly 3, and Bhargava and Ho have shown that, in the family of curves with a marked point, the average is exactly 6. In stark contrast to these results, we show that the average size in the family of elliptic curves with a two-torsion point is unbounded. This follows from an understanding of the Tamagawa ratio associated to such elliptic curves, which we prove is "normally distributed with infinite variance". This work is joint with Zev Klagsbrun. |
Sep 04
Patrick Allen |
Unramified deformation rings |
Class field theory allows one to precisely understand ramification in abelian extensions of number fields. A consequence is that infinite pro-p abelian extensions of a number field are infinitely ramified above p. Boston conjectured a nonabelian analogue of this fact, predicting that certain universal p-adic representations that are unramified at p act via a finite quotient, and this conjecture strengthens the unramified version of the Fontaine-Mazur conjecture. We show in many cases that one can deduce Boston's conjecture from the unramified Fontaine-Mazur conjecture, which allows us to deduce (unconditionally) Boston's conjecture in many two-dimensional cases. This is joint work with F. Calegari. |
Sep 11
Melanie Matchett Wood |
The distribution of sandpile groups of random graphs *** |
The sandpile group is an abelian group associated to a graph, given as the cokernel of the graph Laplacian. An Erdős–Rényi random graph then gives some distribution of random abelian groups. We will give an introduction to various models of random finite abelian groups arising in number theory and the connections to the distribution conjectured by Payne et. al. for sandpile groups. We will talk about the moments of random finite abelian groups, and how in practice these are often more accessible than the distributions themselves, but frustratingly are not a priori guaranteed to determine the distribution. In this case however, we have found the moments of the sandpile groups of random graphs, and proved they determine the measure, and have proven Payne's conjecture. *** This is officially a probability seminar, but will occur in the usual NTS room B105 at a slightly earlier time, 2:25 PM. |
Sep 18
Takehiko Yasuda |
Distributions of rational points and number fields, and height zeta functions |
In this talk, I will talk about my attempt to relate Malle's conjecture on the distribution of number fields with Batyrev and Tschinkel's generalization of Manin's conjecture on the distribution of rational points on singular Fano varieties. The main tool for relating these is the height zeta function. |
Sep 25
Ramin Takloo-Bigash |
Counting orders in number fields and p-adic integrals |
In this talk I will report on a recent work on the distribution of orders in number fields. In particular, I will sketch the proof of an asymptotic formula for the number of orders of bounded discriminant in a given quintic number field emphasizing the role played by p-adic (and motivic) integration. This is joint work with Nathan Kaplan (Yale) and Jake Marcinek (Caltech). |
Oct 02
Pham Huu Tiep |
Nilpotent Hall and abelian Hall subgroups |
To which extent can one generalize the Sylow theorems? One possible direction is to assume the existence of a nilpotent subgroup whose order and index are coprime. We will discuss recent joint work with various collaborators that gives a criterion to detect the existence of such subgroups in any finite group. |
Oct 09
Michael Woodbury |
An Adelic Kuznetsov Trace Formula for GL(4) |
An important tool in analytic number theory for GL(2)-type questions is Kuznetsov’s trace formula. Recently, in work of Blomer and of Goldfeld/Kontorovich, generalizations of this to GL(3) have been given which are useful for number theoretic applications. In my talk I will discuss joint work with Dorian Goldfeld in which we further generalize the said GL(3) results to GL(4). I will discuss some of the new features and complications which arise for GL(4) as well as applications to low lying zeros of L-functions and a vertical Sato-Tate theorem. |
Oct 16
Robert Grizzard |
Small points and free abelian groups |
Let F be an algebraic extension of the rational numbers and E an elliptic curve defined over some number field contained in F. The absolute logarithmic Weil height, respectively the Néron-Tate height, induces a norm on F* modulo torsion, respectively on E(F) modulo torsion. The groups F* and E(F) are free abelian modulo torsion if the height function does not attain arbitrarily small positive values. We prove the failure of the converse to this statement by explicitly constructing counterexamples. This is joint work with Philipp Habegger and Lukas Pottmeyer. |
Oct 23
Tonghai Yang |
A conjecture of Colmez |
In his seminal work on the Mordell conjecture, Faltings introduces and studies the (semistable) height of an Abelian variety. When the Abelian veriety is a CM elliptic curve, its Faltings height is essentially the local derivative (at the critical point s=1) of the Dirichlet L-series associated to the imaginary quadratic field by the famous Chowla-Selberg formula. In 1990s, Colmez gave a precise conjectural formula to compute the Faltings height of a CM abelian variety of CM type (E, Φ) in terms of the log derivative at s=1 of some `Artin' L-function associated to the CM type Φ. He proved the conjecture when the CM number field when E is abelian, refining Gross and Anderson's work on periods. Around 2007, I proved the first non-abelian case of the Colmez conjecture using a totally different method--arithmetic intersection and Borcherds product. In this talk, I will talk about its generalization to a new family of CM type, which is an ongoing joint work with Bruinier, Howard, Kudla, and Rapoport. |
Oct 30
Laura DeMarco |
Elliptic curves and complex dynamics |
I will discuss relations between the dynamics of complex rational functions and the arithmetic of elliptic curves. My goal is to present some new work (in progress) that reproves/generalizes a 1959 result of Lang and Neron about rational points on elliptic curves over function fields. On the dynamical side, the same ideas lead to a characterization of stability for families of rational maps on P^{1}. |
Nov 06
Michael Magee |
Zero sets of Hecke polynomials on the sphere |
The eigenspaces of the Laplacian on the two dimensional sphere consist of homogeneous polynomials and occur with increasing dimension as the eigenvalue grows. I'll explain how one can remove this high multiplicity by using arithmetic Hecke operators that arise from the Hamilton quaternions. The resulting Hecke eigenfunctions are subject to predictions arising from random function theory and quantum chaos, in particular concerning the topology of their zero sets. I'll discuss what is known in this area and how one can try to study these 'nodal lines'. |
Nov 13
Yiwei She |
The Shafarevich conjecture for K3 surfaces |
Let K be a number field, S a finite set of places of K, and g a positive integer. Shafarevich made the following conjecture for higher genus curves: the set of isomorphism classes of genus g curves defined over K and with good reduction outside of S is finite. In 1983, Faltings proved this conjecture for curves and the analogous conjecture for polarized abelian surfaces. Building on the work of Faltings and Andre, we prove the stronger unpolarized Shafarevich conjecture for K3 surfaces. I will also explain the connections between the Shafarevich conjecture and the Tate conjecture. |
Nov 20
Simon Marshall |
Endoscopy and cohomology of unitary groups |
We will give a rough outline of the endoscopic classification of representations of quasi-split unitary groups carried out by Mok, following Arthur and others. We will show how this can be used to prove asymptotics for the L^{2} Betti numbers of families of locally symmetric spaces. |
Dec 04
Joel Specter |
Commuting Endomorphisms of the p-adic Unit Disk |
When can a pair of endomorphisms of [math]\displaystyle{ \mathbf{Z}_p[[X]]/\mathbf{Z}_p }[/math] commute? Approaching this problem from the vantage point of dynamics on the p-adic unit disk, Lubin proved that whenever a non-invertible endomorphism f commutes with a non-torsion automorphism u, the pair f and u exhibit many of the same properties as endomorphisms of a formal group over [math]\displaystyle{ \mathbf{Z}_p }[/math]. Because of this, he posited that for such a pair of endomorphisms to exist, there in fact had to be a formal group 'somehow in the background.' In this talk, I will discuss how some of the dynamical systems of Lubin occur naturally as the restriction of the Galois action on certain Fontaine period rings. Using this observation, I will construct, in some cases, the formal groups conjectured by Lubin. |
Dec 11
Ila Varma |
The mean number of 3-torsion elements in ray class groups of quadratic fields |
In 1971, Davenport and Heilbronn determined the mean number of 3-torsion elements in the class groups of quadratic fields, when ordered by discriminant. I will describe some aspects of the proof of Davenport and Heilbronn’s theorem; in particular, they prove a relationship via class field theory between the number of 3-torsion ideal classes of quadratic fields and the number of nowhere totally ramified cubic fields over [math]\displaystyle{ \mathbb{Q} }[/math]. This argument generalizes to give a relationship between 3-torsion elements of the ray class groups of quadratic fields and certain pairs of cubic fields satisfying explicit ramification conditions. I will illustrate how the combination of this fact with Davenport-Heilbronn’s asymptotics on the number of cubic fields of bounded discriminant allows one to compute the mean number of 3-torsion elements in ray class groups of quadratic fields. If time permits, I will discuss the analogous theorems computing the mean size of 2-torsion elements in ray class groups of cubic fields ordered by discriminant, generalizing Bhargava. |
Organizer contact information
Sean Rostami (srostami@math.wisc.edu)
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