Georgia Institute of Technology College of Sciences

One of the most outstanding problems in differential geometry is concerned with flexibility of closed surface in Euclidean 3-space: Is it possible to continuously deform a smooth closed surface without changing its intrinsic metric structure? In this talk I will give a quick survey of known results in this area, which is primarily concerned with convex surfaces, and outline a program for studying the general case.

Asymptotics for L2 Christoffel functions are a classical topic in orthogonal polynomials. We present asymptotics for Lp Christoffel functions for measures on the unit circle. The formulation involves an extremal problem in Paley-Wiener space. While there have been estimates of the Lp Christoffel functions for a long time, the asymptotics are noew for p other than 2, even for Lebesgue measure on the unict circle.

Self-adjoint $n$-by-$n$ matrices have a natural partial ordering, namely $ A \leq B $ if the matrix $ B - A$ is positive semi-definite. In 1934 K. Loewner characterized functions that preserve this ordering; these functions are called $n$-matrix monotone. The condition depends on the dimension $n$, but if a function is $n$-matrix monotone for all $n$, then it must extend analytically to a function that maps the upper half-plane to itself. I will describe Loewner's results, and then discuss what happens if one wants to characterize functions $f$ of two (or more) variables that are matrix monotone in the following sense: If $ A = (A_1, A_2)$ and $B = (B_1,B_2)$ are pairs of commuting self-adjoint $n$-by-$n$ matrices, with $A_1 \leq B_1 $ and $A_2 \leq B_2$, then $f(A) \leq f (B)$. This talk is based on joint work with Jim Agler and Nicholas Young.

The Gaussian central limit theorem says that for a wide class of stochastic systems, the bell curve (Gaussian distribution) describes the statistics for random fluctuations of important observables. In this talk I will look beyond this class of systems to a collection of probabilistic models which include random growth models, polymers,particle systems, matrices and stochastic PDEs, as well as certain asymptotic problems in combinatorics and representation theory. I will explain in what ways these different examples all fall into a single new universality class with a much richer mathematical structure than that of the Gaussian.

Sampling permutations from S_n is a fundamental problem from probability theory. The nearest neighbor transposition chain M_n is known to converge in time \Theta(n^3 \log n) in the uniform case and time \Theta(n^2) in the constant bias case, in which we put adjacent elements in order with probability p \neq 1/2 and out of order with probability 1-p. In joint work with Prateek Bhakta, Dana Randall and Amanda Streib, we consider the variable bias case where the probability of putting an adjacent pair of elements in order depends on the two elements, and we put adjacent elements x < y in order with probability p_{x,y} and out of order with probability 1-p_{x,y}. The problem of bounding the mixing rate of M_n was posed by Fill and was motivated by the Move-Ahead-One self-organizing list update algorithm. It was conjectured that the chain would always be rapidly mixing if 1/2 \leq p_{x,y} \leq 1 for all x < y, but this was only known in the case of constant bias or when p_{x,y} is equal to 1/2 or 1, a case that corresponds to sampling linear extensions of a partial order. We prove the chain is rapidly mixing for two classes: ``Choose Your Weapon,'' where we are given r_1,..., r_{n-1} with r_i \geq 1/2 and p_{x,y}=r_x for all x < y (so the dominant player chooses the game, thus fixing his or her probability of winning), and ``League Hierarchies,'' where there are two leagues and players from the A-league have a fixed probability of beating players from the B-league, players within each league are similarly divided into sub-leagues with a possibly different fixed probability, and so forth recursively. Both of these classes include permutations with constant bias as a special case. Moreover, we also prove that the most general conjecture is false. We do so by constructing a counterexample where 1/2 \leq p_{x,y} \leq 1 for all x < y, but for which the nearest neighbor transposition chain requires exponential time to converge.

Over the past 40 years, researchers have made many connections between the dimension of posets and the issue of planarity for graphs and diagrams, but there appears to be little work connecting dimension to structural graph theory. This situation has changed dramatically in the last several months. At the Robin Thomas birthday conference, Gwenael Joret, made the following striking conjecture, which has now been turned into a theorem: The dimension of a poset is bounded in terms of its height and the tree-width of its cover graph. In this talk, I will outline how Joret was led to this conjecture by the string of results on planarity. I will also sketch how the resolution of his conjecture points to a number of new problems, which should interest researchers in both communities.