NTS Spring 2012/Abstracts

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 8

Zev Klagsbrun (Madison)

Title: Erdős–Kac Type Theorems

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. The lecture is suitable for general math audience.

March 15

Yongqiang Zhao (Madison)

Title: On the Roberts conjecture

Abstract: Let N(X) = #{K | [K:Q] = 3, disc(K) ≤ X} be the counting function of cubic fields of bounded 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.

March 22

Paul Terwilliger (Madison)

Title: Introduction to tridiagonal pairs

Abstract: tba

March 29

David P. Roberts (U. Minnesota Morris)

Title: Lightly ramified number fields with Galois group S.M12.A

Abstract: Two of the most important invariants of an irreducible polynomial f(x) ∈ Z[x ] are its Galois group G and its field discriminant D. The inverse Galois problem asks one to find a polynomial f(x) having any prescribed Galois group G. Refinements of this problem ask for D to be small in various senses, for example of the form ± pa for the smallest possible prime p. 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 M12, its automorphism group M12.2, its double cover 2.M12, and the combined extension 2.M12.2. Among the polynomials found is f(x) = x48 + 2 e3 x42 + 69 e5 x36 + 868 e7 x30 − 4174 e7 x26 + 11287 e9 x24 − 4174 e10 x20 + 5340 e12 x18 + 131481 e12 x14 +17599 e14 x12 + 530098 e14 x8 + 3910 e16 x6 + 4355569 e14 x4 + 20870 e16 x2 + 729 e18, with e = 11. This polynomial has Galois group G = 2.M12.2 and field discriminant 1188. It makes M12 the first of the twenty-six sporadic simple groups Γ known to have a polynomial with Galois group G involving Γ and field discriminant D the power of single prime dividing |Γ |.

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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