Activities This Week

We consider quasi-orthogonal polynomial families - those that are orthogonal with respect to a non-degenerate bilinear form defined by a linear functional - in which the ratio of successive coefficients is given by a rational function f(n,k) which is polynomial in n. Here, n is the index of the polynomial, and k of the coefficient. We show that, up to rescaling and renormalization, there are only five such families.

More generally, we define an auxiliary basis for the space of polynomials, called Newtonian bases, and consider coefficients with respect to this basis rather than the standard monomial basis. We call the polynomial families that satisfy the rationality conditions on ratio of successive coefficients with respect to this basis HG-families. We show that, up to rescaling, shift, and renormalization, there are only 10 quasi-orthogonal HG-families. Each family arises as a specialization of some hypergeometric series. I will define this notion in the talk. Eight of the 10 families are classical very useful polynomial families, and we view our theorem as a classification result in the theory of special functions.

We also consider the more general rational HG-families, i.e. quasi-orthogonal families in which the ratio f(n,k) of successive coefficients is allowed to be rational in n as well. I will formulate the two main theorems, one on quasi-orthogonal HG-families and one on rational quasi-orthogonal HG-families, as well as the main ideas of the proofs. They are of algebraic nature.

This is a joint work with Joseph Bernstein and Siddhartha Sahi.

פונקציות זיתא מופיעות במקומות רבים במתמטיקה. נתמקד בהרצאה זו במקרה אחד שהוא יחסית קל להבנה. פונקציית הזיתא של איהרה שסופרת מעגלים סגורים בגרף נתון. ננסה להבין מהי המשמעות הגיאומטרית לגבי הגרף לכך שפונקציית הזיתא שלו מקיימת את השערת רימן המתאימה.

In this talk, I will describe some ring theoretic properties of certain rings of noncommutative functions. In particular, I will show that these topological rings are good analogs of the classical rings of analytic functions on discs in the plane. Our rings turn out to be semi-free ideal rings. Namely, every finitely generated right (equivalently, left) ideal is free as a module. In turn, this implies that they admit an embedding into a division ring with a certain universal property (a universal localization). I will explain how this result is a blend of techniques from ring theory and operator algebras and show an application to free probability.

This talk is based on joint work with Meric Augat and Rob Martin.

We will describe a general strategy for proving the algebraicity of the Hodge Weil classes on abelian varieties of Weil type. The latter are even dimensional abelian varieties admitting a suitable embedding of a CM number field in their rational endomorphism ring. We will describe the implementation of the strategy for abelian varieties of dimension 4 and 6, and why it implies the Hodge conjecture for abelian varieties of dimension at most 5.

Let w(x,y) be a word in a free group. For any group G, w induces a word map w:G^2–>G. For example, the commutator word w=xyx^(-1)y^(-1) induces the commutator map. In the setting of finite simple groups, Larsen, Shalev and Tiep showed there exists epsilon(w)>0 (depending only on the word w), such that for all sufficiently large G, the probability that a random pair (g_1,g_2) in G^2 satisfies w(g_1,g_2)=g is smaller than |G|^(-epsilon(w)). They further obtained uniform upper bounds on the L^1- and L^infty-mixing times for the random walks induced by the corresponding word measures.
I will discuss analogous results for the family of unitary groups in all ranks.