Difference between revisions of "NTS Fall 2012/Abstracts"

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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"
 
|-
 
|-
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Who?''' (Where?)
+
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Simon Marshall''' (Northwestern)
 
|-
 
|-
 
| bgcolor="#BCD2EE"  align="center" | Title: Multiplicities of automorphic forms on GL<sub>2</sub>
 
| bgcolor="#BCD2EE"  align="center" | Title: Multiplicities of automorphic forms on GL<sub>2</sub>
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| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Jordan Ellenberg''' (UW–Madison)
 
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Jordan Ellenberg''' (UW–Madison)
 
|-
 
|-
| bgcolor="#BCD2EE"  align="center" | Title: tba
+
| bgcolor="#BCD2EE"  align="center" | Title: Topology of Hurwitz spaces and Cohen-Lenstra conjectures over function fields
 
|-
 
|-
 
| bgcolor="#BCD2EE"  |   
 
| bgcolor="#BCD2EE"  |   
Abstract: tba
+
Abstract: We will discuss recent progress, joint with Akshay Venkatesh and Craig Westerland, towards the Cohen–Lenstra conjecture over the function field '''F'''<sub>''q''</sub>(''t''). There are two key novelties, one topological and one arithmetic. The first is a homotopy-theoretic description of the "moduli space of ''G''-covers with infinitely many branch points." The second is a description of the stable components of Hurwitz space over '''F'''<sub>''q''</sub>, as a module for Gal(<span style="text-decoration:overline;">'''F'''</span><sub>''q''</sub>/'''F'''<sub>''q''</sub>). At least half the talk will be devoted to explaining why these objects are relevant to a very down-to-earth question like Cohen–Lenstra. If time permits, I'll explain what this has to do with the conjectures Nigel spoke about two weeks ago, and a bit about what Daniel is up to.
 
|}                                                                         
 
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<br>
 
<br>
  
<!--
+
== October 4 ==
== February 23 ==
 
  
 
<center>
 
<center>
 
{| 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"
 
|-
 
|-
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Christelle Vincent''' (Madison)
+
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Sean Rostami''' (Madison)
 
|-
 
|-
| bgcolor="#BCD2EE"  align="center" | Title: Drinfeld modular forms
+
| bgcolor="#BCD2EE"  align="center" | Title: Centers of Hecke algebras
 
|-
 
|-
 
| bgcolor="#BCD2EE"  |   
 
| bgcolor="#BCD2EE"  |   
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.
+
Abstract: The classification and construction of smooth
 +
representations of algebraic groups (over non-archimedean local
 +
fields) depends heavily on certain function algebras called Hecke
 +
algebras. The centers of such algebras are particularly important for
 +
classification theorems, and also turn out to be the home of some
 +
trace functions that appear in the Hasse–Weil zeta function of a
 +
Shimura variety. The Bernstein isomorphism is an explicit
 +
identification of the center of an Iwahori–Hecke algebra. I talk about
 +
all these things, and outline a satisfying direct proof of the
 +
Bernstein isomorphism (the theorem is old, the proof is new).
 
|}                                                                         
 
|}                                                                         
 
</center>
 
</center>
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<br>
 
<br>
  
== March 1 ==
+
== October 11 ==
  
 
<center>
 
<center>
 
{| 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"
 
|-
 
|-
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Shamgar Gurevich''' (Madison)
+
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Tonghai Yang''' (Madison)
 
|-
 
|-
| bgcolor="#BCD2EE"  align="center" | Title: Computing the Matched Filter in Linear Time
+
| bgcolor="#BCD2EE"  align="center" | Title: Quaternions and Kudla's matching principle
 
|-
 
|-
 
| bgcolor="#BCD2EE"  |   
 
| bgcolor="#BCD2EE"  |   
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
+
Abstract: In this talk, I will explain some interesting identities among average representation numbers by definite quaternions and degree of Hecke operators on Shimura curves (thus indefinite quaternions).  
 
 
R(t)&nbsp;=&nbsp;exp{2πiωt/''p''}⋅S(t+τ)&nbsp;+&nbsp;W(t),
 
 
 
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.
 
 
 
Problem (digital radar problem) Extract τ,&nbsp;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''<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.
 
 
 
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.
 
  
 
|}                                                                         
 
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<br>
 
<br>
== March 8 ==
+
== October 18 ==
  
 
<center>
 
<center>
 
{| 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"
 
|-
 
|-
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Zev Klagsbrun''' (Madison)
+
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Rachel Davis''' (Madison)
 
|-
 
|-
| bgcolor="#BCD2EE"  align="center" | Title: Erdős–Kac Type Theorems
+
| bgcolor="#BCD2EE"  align="center" | Title: On the images of metabelian Galois representations associated to elliptic curves
 
|-
 
|-
 
| bgcolor="#BCD2EE"  |   
 
| bgcolor="#BCD2EE"  |   
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.
+
Abstract: For ℓ-adic Galois representations associated to elliptic curves, there are theorems
 
+
concerning when the images are surjective. For example, Serre proved that for a fixed non-CM elliptic curve ''E''/'''Q''', for all but finitely many primes ℓ, the ℓ-adic Galois representation is surjective. Grothendieck and others have developed a theory of Galois representations to an automorphism group of a free pro-ℓ group. In this case, there is less known about the size of the images.
The lecture is suitable for general math audience.
+
The goal of this research is to understand Galois representations to automorphism groups of non-abelian groups more tangibly. Let ''E'' be a semistable elliptic curve over '''Q''' of negative discriminant with good supersingular reduction at 2. Associated to ''E'', there is a Galois representation to a subgroup of the automorphism group of a metabelian group. I conjecture this representation is surjective and give evidence for this conjecture. Then, I compute some conjugacy invariants for the images of the Frobenius elements. This will give rise to new arithmetic information analogous to the traces of Frobenius for the ℓ-adic representation.
  
 
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<br>
 
<br>
  
== March 15 ==
+
== October 25 ==
  
 
<center>
 
<center>
 
{| 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"
 
|-
 
|-
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Yongqiang Zhao''' (Madison)
+
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Shamgar Gurevich''' (Madison)
 
|-
 
|-
| bgcolor="#BCD2EE"  align="center" | Title: On the Roberts conjecture
+
| bgcolor="#BCD2EE"  align="center" | Title: Bounds on characters of SL(2,&nbsp;''q'') via Theta correspondence
 
|-
 
|-
 
| bgcolor="#BCD2EE"  |   
 
| bgcolor="#BCD2EE"  |   
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
+
Abstract: I will report on part from a joint project with Roger Howe (Yale).  We develop a method to obtain effective bounds on the irreducible characters of SL(2,&nbsp;''q''). Our method uses explicit realization of all the irreducible representations via the Theta correspondence applied to the dual pair (SL(2,&nbsp;''q''),&nbsp;O), where O is an orthogonal group.
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
 
approaches to this problem were developed independently by Bhargava, Shankar and Tsimerman,
 
Hough, Taniguchi and Thorne, and myself.
 
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
 
feeds back to the number field case, in particular, how one could possibly define a new invariant
 
for cubic fields.
 
  
 +
If you want to learn what are all the notions in my abstract you are welcome to attend the talk. I will not assume any familiarity with the subject.
 
|}                                                                         
 
|}                                                                         
 
</center>
 
</center>
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<br>
 
<br>
  
== March 22 ==
+
== November 1 ==
  
 
<center>
 
<center>
 
{| 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"
 
|-
 
|-
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Paul Terwilliger''' (Madison)
+
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Lei Zhang''' (Boston College)
 
|-
 
|-
| bgcolor="#BCD2EE"  align="center" | Title: Introduction to tridiagonal pairs
+
| bgcolor="#BCD2EE"  align="center" | Title: Tensor product L-functions of classical groups of Hermitian type: quasi-split case
 
|-
 
|-
 
| bgcolor="#BCD2EE"  |   
 
| bgcolor="#BCD2EE"  |   
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.
+
Abstract: We study the tensor product L-functions for quasi-split classical groups of Hermitian type times general linear group. When the irreducible automorphic representation of the general linear group is cuspidal and the classical group is an orthogonal group, the tensor product L-functions have been studied by Ginzburg, Piatetski-Shapiro and Rallis in 1997. In this talk, we will show that the global integrals are eulerian and finish the explicit calculation of unramified local L-factors in general.
 +
|}                                                                       
 +
</center>
  
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;
 
#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.
 
  
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.
+
== November 8 ==
  
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.
+
<center>
 +
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
 +
|-
 +
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Robert Harron''' (UW–Madison)
 +
|-
 +
| bgcolor="#BCD2EE"  align="center" | Title: Computing Hida families
 +
|-
 +
| bgcolor="#BCD2EE"  | 
 +
Abstract: I will report on a joint project with Rob Pollack and four people you know well: Evan Dummit,
 +
Marton Hablicsek, Lalit Jain, and Daniel Ross. Our goal is to explicitly compute
 +
Hida families using overconvergent modular symbols. This grew out of a
 +
project at the Arizona Winter School and the basic idea is to study
 +
''p''-adic families of overconvergent modular symbols. I will go over the
 +
basic definitions and results starting from classical modular symbols and explain
 +
how one goes about encoding these objects on a computer. Aside from
 +
being able to compute formal ''q''-expansions of Hida families, we can
 +
also compute the structure of the ordinary ''p''-adic Hecke algebra,
 +
''L''-invariants, two-variable ''p''-adic ''L''-functions, etc. Several examples
 +
will be provided. The code is implemented in Sage.
 +
|}                                                                       
 +
</center>
  
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>
 
<li>Each of ''A'', ''A''<sup>∗</sup> is diagonalizable on ''V'';
 
<li>There exists an ordering {''V<sub>i</sub>''}<sub>''i''=0,...,''d''</sub> of the eigenspaces of ''A'' such that
 
::''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''),
 
where ''V''<sub>&minus;1</sub>&nbsp;=&nbsp;0, ''V''<sub>''d''+1</sub>&nbsp;=&nbsp;0;
 
<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
 
::''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;),
 
where ''V''<sup>*</sup><sub>&minus;1</sub>&nbsp;=&nbsp;0, ''V''<sup>*</sup><sub>''d''+1</sub>&nbsp;=&nbsp;0;
 
<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''.
 
</ol>
 
It turns out that ''d''&nbsp;=&nbsp;δ and this common value is called the diameter of the pair.
 
A Leonard pair is the same thing as a tridiagonal pair for which the eigenspaces
 
''V'' and ''V''<sup>∗</sup> all have dimension 1.
 
  
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.
+
== November 15 ==
  
In this talk we will summarize the basic facts about a tridiagonal pair, describing
+
<center>
features such as the eigenvalues, dual eigenvalues, shape, tridiagonal relations,
+
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
split decomposition, and parameter array. We will then focus on a special case
+
|-
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.
+
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Robert Lemke Oliver''' (Emory)
 
+
|-
In our main result we classify the sharp tridiagonal pairs up to isomorphism.
+
| bgcolor="#BCD2EE"  align="center" | Title: Multiplicative functions with small sums
 +
|-
 +
| bgcolor="#BCD2EE"  | 
 +
Abstract: Analytic number theory is in need of new ideas: for the very
 +
problem which motivated its existence – the distribution of primes – we
 +
have been unable to make progress in more than fifty years.  Granville and
 +
Soundararajan have recently put forward a possible substitute for the
 +
seemingly intractable, though admittedly rich, theory of zeros of
 +
''L''-functions.  They dub this new framework the pretentious view of analytic
 +
number theory, where the main objects of consideration are generic
 +
multiplicative functions, and the goal is to obtain deep theorems about the
 +
structure of the partial sums of such functions.  In this talk, we consider
 +
multiplicative functions whose partial sums exhibit extreme cancellation.
 +
We will present two different lines of work about this problem. First, we
 +
develop what might be considered the pretentious framework to answer this
 +
question – notions of pretentious which permit the detection of power
 +
cancellation – which is joint work with Junehyuk Jung of Princeton
 +
University. Second, we consider a natural class of functions defined via
 +
the arithmetic of number fields, and we classify the members of this class
 +
which exhibit extreme cancellation; the proof of this is not at all
 +
pretentious.  
 
|}                                                                         
 
|}                                                                         
 
</center>
 
</center>
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<br>
 
<br>
  
== March 29 ==
+
== November 29 ==
  
 
<center>
 
<center>
 
{| 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"
 
|-
 
|-
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''David P. Roberts''' (U. Minnesota Morris)
+
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Xin Shen''' (Minnesota)
 
|-
 
|-
| bgcolor="#BCD2EE"  align="center" | Title: Lightly ramified number fields with Galois group ''S''.''M''<sub>12</sub>.''A''
+
| bgcolor="#BCD2EE"  align="center" | Title: Unramified computation for automorphic tensor ''L''-function
 
|-
 
|-
 
| bgcolor="#BCD2EE"  |
 
| bgcolor="#BCD2EE"  |
Abstract: Two of the most important invariants of an irreducible polynomial ''f''(''x'')&nbsp;&isin;&nbsp;'''Z'''[''x''&thinsp;] are its
+
Abstract: In 1967 Langlands introduced the automorphic ''L''-functions and conjectured their analytic properties, including the meromorphic continuation to
Galois group ''G'' and its field discriminant ''D''.    The inverse Galois problem asks
+
'''C''' with finitely many poles and a standard functional equation. One
one to find a polynomial ''f''(''x'') having any prescribed Galois group ''G''.  Refinements
+
of the important cases is the tensor ''L''-functions for ''G''&nbsp;&times;&nbsp;GL<sub>''k''</sub> where
of this problem ask for ''D'' to be small in various senses, for example of the form
+
''G'' is a classical group. In this seminar I will survey some approaches to this case via integral representations. I will also talk about my recent
&plusmn;&thinsp;''p<sup>a''</sup> for the smallest possible prime ''p''
+
work on the unramified computation for ''L''-functions of Sp<sub>2''n''</sub>&nbsp;&times;&nbsp;GL<sub>''k''</sub>
 
+
for the non-generic case.
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
 
{| style="background: #BCD2EE;" align="center"
 
|-
 
| ''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>
 
|-
 
| ||&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>
 
|-
 
| ||+ 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''&nbsp;=&nbsp;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 &Gamma;
 
known to have a polynomial with Galois group
 
''G'' involving &Gamma; and field discriminant ''D''
 
the power of a single prime dividing |&Gamma;&thinsp;|.
 
 
 
 
|}                                                                         
 
|}                                                                         
 
</center>
 
</center>
 
  
 
<br>
 
<br>
  
== April 12 ==
+
== December 6 ==
  
 
<center>
 
<center>
 
{| 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"
 
|-
 
|-
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Chenyan Wu''' (Minnesota)
+
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Sheng-Chi Liu''' (Texas A&M)
 
|-
 
|-
| bgcolor="#BCD2EE"  align="center" | Title: Rallis inner product formula for theta lifts from metaplectic groups to orthogonal groups
+
| bgcolor="#BCD2EE"  align="center" | Title: Subconvexity and equidistribution of Heegner points in the level aspect
 
|-
 
|-
 
| bgcolor="#BCD2EE"  |   
 
| bgcolor="#BCD2EE"  |   
Abstract: Let π be a genuine cuspidal representation of the metaplectic group of rank
+
Abstract: We will discuss the equidistribution property of Heegner points of level ''q'' and discriminant ''D'', as ''q'' and ''D'' go to infinity. We will establish a hybrid subconvexity bound for certain Rankin–Selberg ''L''-functions which are related to the equdistribution of Heegner points. This is joint work with Riad Masri and Matt Young.  
''n''. We consider the theta lifts to the orthogonal group associated to a
 
quadratic space of dimension 2''n''&nbsp;+&nbsp;1. We show a case of a regularized Rallis inner
 
product formula that relates the pairing of theta lifts to the central value of the
 
Langlands ''L''-function of π twisted by a genuine character. This enables us to
 
demonstrate the relation between non-vanishing of theta lifts and the non-vanishing of
 
central ''L''-values. We prove also a case of regularized Siegel–Weil formula which is
 
missing in the literature, as it forms the basis of our proof of the Rallis inner
 
product formula.
 
  
 
|}                                                                         
 
|}                                                                         
 
</center>
 
</center>
 
+
<!--
 
<br>
 
<br>
  

Latest revision as of 22:45, 9 November 2012

September 13

Nigel Boston (UW–Madison)
Title: Non-abelian Cohen–Lenstra heuristics

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 group A is isomorphic to the Galois group of the maximal unramified abelian p-extension of K. In work with Michael Bush and Farshid Hajir, I generalized this to non-abelian unramified p-extensions of imaginary quadratic fields. I shall recall all the above and describe a further generalization to non-abelian unramified p-extensions of H-extensions of Q, for any p, H, where p does not divide the order of H.


September 20

Simon Marshall (Northwestern)
Title: Multiplicities of automorphic forms on GL2

Abstract: I will discuss some ideas related to the theory of p-adically completed cohomology developed by Frank Calegari and Matthew Emerton. If F is a number field which is not totally real, I will use these ideas to prove a strong upper bound for the dimension of the space of cohomological automorphic forms on GL2 over F which have fixed level and growing weight.


September 27

Jordan Ellenberg (UW–Madison)
Title: Topology of Hurwitz spaces and Cohen-Lenstra conjectures over function fields

Abstract: We will discuss recent progress, joint with Akshay Venkatesh and Craig Westerland, towards the Cohen–Lenstra conjecture over the function field Fq(t). There are two key novelties, one topological and one arithmetic. The first is a homotopy-theoretic description of the "moduli space of G-covers with infinitely many branch points." The second is a description of the stable components of Hurwitz space over Fq, as a module for Gal(Fq/Fq). At least half the talk will be devoted to explaining why these objects are relevant to a very down-to-earth question like Cohen–Lenstra. If time permits, I'll explain what this has to do with the conjectures Nigel spoke about two weeks ago, and a bit about what Daniel is up to.


October 4

Sean Rostami (Madison)
Title: Centers of Hecke algebras

Abstract: The classification and construction of smooth representations of algebraic groups (over non-archimedean local fields) depends heavily on certain function algebras called Hecke algebras. The centers of such algebras are particularly important for classification theorems, and also turn out to be the home of some trace functions that appear in the Hasse–Weil zeta function of a Shimura variety. The Bernstein isomorphism is an explicit identification of the center of an Iwahori–Hecke algebra. I talk about all these things, and outline a satisfying direct proof of the Bernstein isomorphism (the theorem is old, the proof is new).


October 11

Tonghai Yang (Madison)
Title: Quaternions and Kudla's matching principle

Abstract: In this talk, I will explain some interesting identities among average representation numbers by definite quaternions and degree of Hecke operators on Shimura curves (thus indefinite quaternions).


October 18

Rachel Davis (Madison)
Title: On the images of metabelian Galois representations associated to elliptic curves

Abstract: For ℓ-adic Galois representations associated to elliptic curves, there are theorems concerning when the images are surjective. For example, Serre proved that for a fixed non-CM elliptic curve E/Q, for all but finitely many primes ℓ, the ℓ-adic Galois representation is surjective. Grothendieck and others have developed a theory of Galois representations to an automorphism group of a free pro-ℓ group. In this case, there is less known about the size of the images. The goal of this research is to understand Galois representations to automorphism groups of non-abelian groups more tangibly. Let E be a semistable elliptic curve over Q of negative discriminant with good supersingular reduction at 2. Associated to E, there is a Galois representation to a subgroup of the automorphism group of a metabelian group. I conjecture this representation is surjective and give evidence for this conjecture. Then, I compute some conjugacy invariants for the images of the Frobenius elements. This will give rise to new arithmetic information analogous to the traces of Frobenius for the ℓ-adic representation.


October 25

Shamgar Gurevich (Madison)
Title: Bounds on characters of SL(2, q) via Theta correspondence

Abstract: I will report on part from a joint project with Roger Howe (Yale). We develop a method to obtain effective bounds on the irreducible characters of SL(2, q). Our method uses explicit realization of all the irreducible representations via the Theta correspondence applied to the dual pair (SL(2, q), O), where O is an orthogonal group.

If you want to learn what are all the notions in my abstract you are welcome to attend the talk. I will not assume any familiarity with the subject.


November 1

Lei Zhang (Boston College)
Title: Tensor product L-functions of classical groups of Hermitian type: quasi-split case

Abstract: We study the tensor product L-functions for quasi-split classical groups of Hermitian type times general linear group. When the irreducible automorphic representation of the general linear group is cuspidal and the classical group is an orthogonal group, the tensor product L-functions have been studied by Ginzburg, Piatetski-Shapiro and Rallis in 1997. In this talk, we will show that the global integrals are eulerian and finish the explicit calculation of unramified local L-factors in general.


November 8

Robert Harron (UW–Madison)
Title: Computing Hida families

Abstract: I will report on a joint project with Rob Pollack and four people you know well: Evan Dummit, Marton Hablicsek, Lalit Jain, and Daniel Ross. Our goal is to explicitly compute Hida families using overconvergent modular symbols. This grew out of a project at the Arizona Winter School and the basic idea is to study p-adic families of overconvergent modular symbols. I will go over the basic definitions and results starting from classical modular symbols and explain how one goes about encoding these objects on a computer. Aside from being able to compute formal q-expansions of Hida families, we can also compute the structure of the ordinary p-adic Hecke algebra, L-invariants, two-variable p-adic L-functions, etc. Several examples will be provided. The code is implemented in Sage.


November 15

Robert Lemke Oliver (Emory)
Title: Multiplicative functions with small sums

Abstract: Analytic number theory is in need of new ideas: for the very problem which motivated its existence – the distribution of primes – we have been unable to make progress in more than fifty years. Granville and Soundararajan have recently put forward a possible substitute for the seemingly intractable, though admittedly rich, theory of zeros of L-functions. They dub this new framework the pretentious view of analytic number theory, where the main objects of consideration are generic multiplicative functions, and the goal is to obtain deep theorems about the structure of the partial sums of such functions. In this talk, we consider multiplicative functions whose partial sums exhibit extreme cancellation. We will present two different lines of work about this problem. First, we develop what might be considered the pretentious framework to answer this question – notions of pretentious which permit the detection of power cancellation – which is joint work with Junehyuk Jung of Princeton University. Second, we consider a natural class of functions defined via the arithmetic of number fields, and we classify the members of this class which exhibit extreme cancellation; the proof of this is not at all pretentious.


November 29

Xin Shen (Minnesota)
Title: Unramified computation for automorphic tensor L-function

Abstract: In 1967 Langlands introduced the automorphic L-functions and conjectured their analytic properties, including the meromorphic continuation to C with finitely many poles and a standard functional equation. One of the important cases is the tensor L-functions for G × GLk where G is a classical group. In this seminar I will survey some approaches to this case via integral representations. I will also talk about my recent work on the unramified computation for L-functions of Sp2n × GLk for the non-generic case.


December 6

Sheng-Chi Liu (Texas A&M)
Title: Subconvexity and equidistribution of Heegner points in the level aspect

Abstract: We will discuss the equidistribution property of Heegner points of level q and discriminant D, as q and D go to infinity. We will establish a hybrid subconvexity bound for certain Rankin–Selberg L-functions which are related to the equdistribution of Heegner points. This is joint work with Riad Masri and Matt Young.


Organizer contact information

Robert Harron

Zev Klagsbrun

Sean Rostami


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