Difference between revisions of "NTS Spring 2012/Abstracts"

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Abstract: Wiles, Taylor, Harris and others used the notation of a big
+
Abstract: Wiles, Taylor, Harris and others used the notion of a big
 
representation of a finite
 
representation of a finite
group to show that certain representations are automorphic.  Thorne
+
group to show that certain representations are automorphic.  Jack Thorne
 
recently observed
 
recently observed
 
that one could weaken this notion of bigness to get the same conclusions.  He
 
that one could weaken this notion of bigness to get the same conclusions.  He
called this property adequate.  An absolutely irreducible  representation V
+
called this property adequate.  An absolutely irreducible  representation ''V''
of a finite group G in characteristic p  is called  adequate  if G has
+
of a finite group ''G'' in characteristic ''p'' is called  adequate  if ''G'' has
no p-quotients, the dimension
+
no ''p''-quotients, the dimension
of V is prime to p,  V has non-trivial self extensions and End(V) is
+
of ''V'' is prime to ''p''''V'' has non-trivial self extensions and End(''V'') is
 
generated by the linear
 
generated by the linear
span of the elements of order prime to p in G.    If G has order
+
span of the elements of order prime to ''p'' in ''G''.    If ''G'' has order
prime to p, all of these conditions
+
prime to ''p'', all of these conditions
hold -- the last condition is sometimes called Burnside's Lemma.  We
+
hold—the last condition is sometimes called Burnside's Lemma.  We
 
will discuss a recent
 
will discuss a recent
result of Guralnick, Herzig, Taylor and Thorne showing that if  p >
+
result of Guralnick, Herzig, Taylor and Thorne showing that if  ''p'' >
2 dim V + 2, then
+
2 dim ''V'' + 2, then
 
any absolutely irreducible representation is adequate.  We will also
 
any absolutely irreducible representation is adequate.  We will also
 
discuss some examples
 
discuss some examples
showing that the span of the p'-elements in End(V) need not be all of End(V).
+
showing that the span of the ''p'''-elements in End(''V'') need not be all of End(''V'').
 
 
 
 
 
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Revision as of 11:12, 27 February 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 Fq[[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 Fq[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 p2⋅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 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: tba

Abstract: tba



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

Shamgar Gurevich

Robert Harron

Zev Klagsbrun

Melanie Matchett Wood



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