NTS Fall 2012/Abstracts: Difference between revisions

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== October 11 ==


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| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Shamgar Gurevich''' (Madison)
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Tonghai Yang''' (Madison)
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| bgcolor="#BCD2EE"  align="center" | Title: Computing the Matched Filter in Linear Time
| bgcolor="#BCD2EE"  align="center" | Title: Quaternions and Kudla's matching principle
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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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== March 8 ==
== October 18 ==


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| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Zev Klagsbrun''' (Madison)
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Rachel Davis''' (Madison)
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| 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
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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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Revision as of 21:21, 10 October 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.



Organizer contact information

Robert Harron

Zev Klagsbrun

Sean Rostami

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