NTS Spring 2012/Abstracts

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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 F_q[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: tba

Abstract: tba

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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