NTS Spring 2012/Abstracts: Difference between revisions
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Abstract:  Abstract: Wiles, Taylor, Harris and others used the notation of a big  
representation of a finite  
group to show that certain representations are automorphic. 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 pquotients, the dimension  
of V is prime to p, V has nontrivial 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).  
}  } 
Revision as of 15:16, 26 February 2012
February 2
Evan Dummit (Madison) 
Title: Kakeya sets over nonarchimedean local rings 
Abstract: In a forthcoming paper with Marci Habliscek, we constructed a Kakeya set over the formal power series ring F_{q}[[t ]], answering a question posed by Ellenberg, Oberlin, and Tao. My talk will be devoted to explaining some of the older history of the Kakeya problem in analysis and the newer history of the Kakeya problem in combinatorics, including my joint work with Marci. In particular, I will give Dvir's solution of the Kakeya problem over finite fields, and explain the problem's extension to other classes of rings. 
February 16
Tonghai Yang (Madison) 
Title: A little linear algebra on CM abelian surfaces 
Abstract: In this talk, I will discuss an interesting Hermitian form structure on the space of special endormorphisms of a CM abelian surface, and how to use it make a moduli problem and prove an arithmetic Siegel–Weil formula over a real quadratic field. This is a joint work with Ben Howard. 
February 23
Christelle Vincent (Madison) 
Title: Drinfeld modular forms 
Abstract: We will begin by introducing the Drinfeld setting, and in particular Drinfeld modular forms and their connection to the geometry of Drinfeld modular curves. We will then present some results about Drinfeld modular forms that we obtained in the process of computing certain geometric points on Drinfeld modular curves. More precisely, we will talk about Drinfeld modular forms modulo P, for P a prime ideal in F_{q}[T ], and about Drinfeld quasimodular forms. 
March 1
Shamgar Gurevich (Madison) 
Title: Computing the Matched Filter in Linear Time 
Abstract: In the digital radar problem we design a function (waveform) S(t) in the Hilbert space H = C(Z/p) of complex valued functions on Z/p = {0, ..., p − 1}, the integers modulo a prime number p ≫ 0. We transmit the function S(t) using the radar to the object that we want to detect. The wave S(t) hits the object, and is reflected back via the echo wave R(t) in H, which has the form R(t) = exp{2πiωt/p}⋅S(t+τ) + W(t), where W(t) in H is a white noise, and τ, ω in Z/p, encode the distance from, and velocity of, the object. Problem (digital radar problem) Extract τ, ω from R and S. In the lecture I first introduce the classical matched filter (MF) algorithm that suggests the 'traditional' way (using fast Fourier transform) to solve the digital radar problem in order of p^{2}⋅log(p) operations. I will then explain how to use techniques from group representation theory and arithmetic to design (construct) waveforms S(t) which enable us to introduce a fast matched filter (FMF) algorithm, that we call the "flag algorithm", which solves the digital radar problem in a much faster way of order of p⋅log(p) operations. I will demonstrate additional applications to mobile communication, and global positioning system (GPS). This is a joint work with A. Fish (Math, Madison), R. Hadani (Math, Austin), A. Sayeed (Electrical Engineering, Madison), and O. Schwartz (Electrical Engineering and Computer Science, Berkeley). The lecture is suitable for general math/engineering audience. 
March 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: tba 
Abstract: Wiles, Taylor, Harris and others used the notation of a big representation of a finite group to show that certain representations are automorphic. 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 pquotients, the dimension of V is prime to p, V has nontrivial 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
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