Events for the week of May 21 2012

Abstract. There is a long history on the study of zeros of random polynomials whose coefficients are independent, identically distributed, non-degenerate random variables. We will first provide an overview on zeros of random functions and then show exact and/or asymptotic bounds on probabilities that all zeros of a random polynomial are real under various distributions. The talk is accessible to undergraduate and graduate students in any areas of mathematics.

Abstract. We will prove a recent theorem of Hida which asserts, under mild conditions, fullness of the big Galois representation associated to an ordinary non-CM family of modular forms.

Abstract. This talk will be a progress report on an attempt to give algebraic versions of Waldspurger's results on the theta correspondence by using Mumford's theory of algebraic theta functions. While the basic machinery can be established, there are many questions still unresolved and many interesting open questions.

Abstract. The Catalan numbers 1, 2, 5, 14, 42, ... are known to count many mathematical objects. Some of the more well-known objects include triangulations of an $n+2$-gon or ways of closing up $n$ pairs of parentheses. (See Richard Stanley's ``Enumerative Combinatorics" for a list of over 150 different combinatorial interpretations.)

We will consider two such families of objects that both carry a natural action of a parbolic quotient of the affine symmetric group: dominant regions in the Shi arrangement and partitions that are both $n$-cores and $n+1$-cores (these will be defined in the talk). While cores are not as ubiquitous as the Catalan numbers, they do arise in several interesting settings. In joint work with Susanna Fishel, we give a bijective proof that these objects are counted by Catalan numbers, (given necessary definitions along the way) using the techniques of J. Anderson.

The talk will be accessible to a general audience.

Abstract. We'll discuss some more or less well-known results on automorphic forms on SL(2), and give some arithmetic consequences (which may be less well-known). The main idea is to explain how some familiar questions from number theory maybe very naturally formulated in terms of automorphic forms on SL(2), rather than the usual group GL(2), and exhibit some curious coincidences which we cannot explain. In conclusion, we will state and prove a converse theorem for the group SL(2).

Abstract. It has been observed by several authors that a large family of stochastic partial differential equations have solutions that are highly intermittent; this means that the solution tends to develop tall peaks that are distributed over small sets ["islands"]. In this talk we will argue that the solution of such stochastic PDEs can be "chaotic," and that this property leads to the onset of intermittence.

This talk is based on joint works with Daniel Conus, Mathew Joseph, and Shang-Yuan Shiu.

Abstract. For any $C^m$-function $g: \R^n \rightarrow \R$, we can consider it as a family of continuous functions $(D^\alpha g)_{\alpha \in \N^n, \left|\alpha\right|\leq m}$ called a jet of order $m$ of $g$. In 1934, Whitney gave a necessary and sufficient condition to determine whether a collection $(f^\alpha)_{\alpha \in \N^d,\left|\alpha\right| \leq m}$ of continuous functions on closed subset $E \subseteq \R^n$ can be extended to a jet of order $m$ of some $C^m$-function. In this talk, we will discuss this problem in o-minimal structures.

Abstract. We consider an integral operatot T with kernel K on a measure space. We assume the kernel in non-negative, symmetric, and satisfies a quasi-metric condition (for example, the Riesz kernel). We obtain matching upper and lower bounds (with different constants) for the kernel of the Neumann series T + T2 + T3 + ..., where the upper bound requires that the operator T be L2 bounded with norm less than 1.

As an application, we obtain estimates for Green's functions of Schrodinger operators, either on all of R^n or on domains, and for the Feynmna-Kac gauge on domains. Our methods apply also when the Laplacian is replaced by a fractional power of the Laplacian.

Abstract. Abstract: In this talk, I will discuss an ongoing attempt to understand the unitary representations of discrete countable groups from the point of view of descriptive set theory. In particular, I will explain why there are precisely two unitary duals of countably infinite amenable groups up to Borel isomorphism.