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 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: In this talk we consider a linear algebraic object called a tridiagonal pair. This object originated in algebraic graph theory, and has connections to orthogonal polynomials and representation theory. In our discussion we aim at a general mathematical audience; no prior experience with the above topics is assumed. Our main results are joint work with Tatsuro Ito and Kazumasa Nomura.

The concept of a tridiagonal pair is best explained by starting with a special case called a Leonard pair. Let F denote a field, and let V denote a vector space over F with finite positive dimension. By a Leonard pair on V we mean a pair of linear transformations A : V → V and A : V → V that satisfy the following two conditions:

  1. There exists a basis for V with respect to which the matrix representing A is diagonal and the matrix representing A is irreducible tridiagonal;
  2. There exists a basis for V with respect to which the matrix representing A is diagonal and the matrix representing A is irreducible tridiagonal.

We recall that a tridiagonal matrix is irreducible whenever each entry on the superdiagonal is nonzero and each entry on the subdiagonal is nonzero. The name "Leonard pair" is motivated by a connection to a 1982 theorem of the combinatorialist Doug Leonard involving the q-Racah and related polynomials of the Askey scheme. Leonard’s theorem was heavily influenced by the work of Eiichi Bannai and Tatsuro Ito on P- and Q- polynomial association schemes, and the work of Richard Askey on orthogonal polynomials. These works in turn were influenced by the work of Philippe Delsarte on coding theory, dating from around 1973.

The central result about Leonard pairs is that they are in bijection with the orthogonal polynomials that make up the terminating branch of the Askey scheme. This branch consists of the q-Racah, q-Hahn, dual q-Hahn, q-Krawtchouk, dual q-Krawtchouk, quantum q-Krawtchouk, affine q-Krawtchouk, Racah, Hahn, dual-Hahn, Krawtchouk, and the Bannai/Ito polynomials. The bijection makes it possible to develop a uniform theory of these polynomials starting from the Leonard pair axiom, and we have done this over the past several years.

A tridiagonal pair is a generalization of a Leonard pair and defined as follows. Let V denote a vector space over F with finite positive dimension. A tridiagonal pair on V is a pair of linear transformations A : V → V and A : V → V that satisfy the following four conditions:

  1. Each of A, A is diagonalizable on V;
  2. There exists an ordering {Vi}i=0,...,d of the eigenspaces of A such that
    A*Vi ⊆ Vi−1 + Vi + Vi+1  (0 ≤ i ≤ d),
    where V−1 = 0, Vd+1 = 0;
  3. There exists an ordering {Vi*}i=0,...,δ of the eigenspaces of A* such that
    AVi* ⊆ V*i−1 + V*i + V*i+1  (0 ≤ i ≤ δ),
    where V*−1 = 0, V*d+1 = 0;
  4. There is no subspace W ⊆ V such that AW ⊆ W, A*W ⊆ W, W ≠ 0, W ≠ V.

It turns out that d = δ and this common value is called the diameter of the pair. A Leonard pair is the same thing as a tridiagonal pair for which the eigenspaces V and V all have dimension 1.

Tridiagonal pairs arise naturally in the theory of P- and Q-polynomial association schemes, in connection with irreducible modules for the subconstituent algebra. They also appear in recent work of Pascal Baseilhac on the Ising model and related structures in statistical mechanics.

In this talk we will summarize the basic facts about a tridiagonal pair, describing features such as the eigenvalues, dual eigenvalues, shape, tridiagonal relations, split decomposition, and parameter array. We will then focus on a special case said to be sharp and defined as follows. Referring to the tridiagonal pair A, A in the above definition, it turns out that for 0 ≤ i ≤ d the dimensions of Vi, Vdi, V*i, V*di coincide; the pair A, A is called sharp whenever V0 has dimension 1. It is known that if F is algebraically closed then A, A is sharp.

In our main result we classify the sharp tridiagonal pairs up to isomorphism.

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 a single prime dividing |Γ |.

April 12

Chenyan Wu (Minnesota)
Title: Rallis inner product formula for theta lifts from metaplectic groups to orthogonal groups

Abstract: Let π be a genuine cuspidal representation of the metaplectic group of rank n. We consider the theta lifts to the orthogonal group associated to a quadratic space of dimension 2n + 1. We show a case of a regularized Rallis inner product formula that relates the pairing of theta lifts to the central value of the Langlands L-function of π twisted by a genuine character. This enables us to demonstrate the relation between non-vanishing of theta lifts and the non-vanishing of central L-values. We prove also a case of regularized Siegel–Weil formula which is missing in the literature, as it forms the basis of our proof of the Rallis inner product formula.

April 16 (special day: Monday, special time: 3:30pm–4:30pm, special place: VV B139)

Hourong Qin (Nanjing U., China)
Title: CM elliptic curves and quadratic polynomials representing primes

Abstract: No polynomial of degree two or higher has been proved to represent infinitely many primes. Let E be an elliptic curve defined over Q with complex multiplication. Fix an integer r. We give sufficient and necessary conditions for ap = r for some prime p. We show that there are infinitely many primes p such that ap = r for some fixed integer r if and only if a quadratic polynomial represents infinitely many primes p.

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: Secondary terms in counting functions for cubic fields

Abstract: We will discuss our proof of secondary terms of order X5/6 in the Davenport–Heilbronn theorems on cubic fields and 3-torsion in class groups of quadratic fields. For cubic fields this confirms a conjecture of Datskovsky–Wright and Roberts. We also will describe some generalizations, in particular to arithmetic progressions, where we discover a curious bias in the secondary term.

Roberts’ conjecture has also been proved independently by Bhargava, Shankar, and Tsimerman. Their proof uses the geometry of numbers, while our proof uses the analytic theory of Shintani zeta functions.

We will also discuss a combined approach which yields further improved error terms. If there is time (or after the talk), I will also discuss a couple of side projects and my plans for further related work.

This is joint work with Takashi Taniguchi.

May 3

Alina Cojocaru (U. Illinois at Chicago)
Title: Frobenius fields for elliptic curves

Abstract: Let E be an elliptic curve defined over Q. For a prime p of good reduction for E, let πp be the p-Weil root of E and Qp) the associated imaginary quadratic field generated by πp. In 1976, Serge Lang and Hale Trotter formulated a conjectural asymptotic formula for the number of primes p < x for which Qp) is isomorphic to a fixed imaginary quadratic field. I will discuss progress on this conjecture, in particular an average result confirming the predicted asymptotic formula. This is joint work with Henryk Iwaniec and Nathan Jones.

May 10

Samit Dasgupta (UC Santa Cruz)
Title: The p-adic L-functions of evil Eisenstein series

Abstract: Let f be a newform of weight k+2 on Γ1(N), and let p ∤ N be a prime. For each root α of the Hecke polynomial of f at p, there is a corresponding p-stabilization fα on Γ1(N) ∩ Γ0(p) with Up-eigenvalue equal to α. The construction of p-adic L-functions associated to such forms fα has been much studied. The non-critical case (when ordp(α) < k+1) was handled in the 1970s via interpolation of the classical L-function in work of Mazur, Swinnerton-Dyer, Manin, Visik, and Amice–Vélu. Recently, certain critical cases were handled by Pollack and Stevens, and the remaining cases were finished off by Bellaïche. Many years prior to Bellaïche's proof of their existence, Stevens had conjectured a factorization formula for the p-adic L-functions of evil (i.e. critical) Eisenstein series based on computational evidence. In this talk we describe a proof of Stevens's factorization formula. A key element of the proof is the theory of distribution-valued partial modular symbols. This is joint work with Joël Bellaïche.

Organizer contact information

Shamgar Gurevich

Robert Harron

Zev Klagsbrun

Melanie Matchett Wood

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