Mathematics

K-theory computations for topological insulators

Aaron Royer (UT Austin)

Abstract: Over the past decade a new class materials called topological insulators, often with counter-intuitive electric properties, have been discovered. The mathematics involved in classifying such materials is twisted equivariant Real K-theory. I will briefly describe the setup and survey new K-theory computations coming from these considerations. This is joint work with Dan Freed.

Constructing solutions of Hitchin's equations near the ends of the moduli space

Laura Fredrickson (University of Texas - Austin)

Abstract: Hitchin's equations are a system of gauge theoretic equations on a Riemann surface that are of interest in many areas including representation theory, Teichmuller theory, and the geometric Langlands correspondence. In this talk, I'll describe what solutions of SL(n,C)-Hitchin's equations ``near the ends'' of the moduli space look like. This construction generalizes Mazzeo-Swoboda-Weiss-Witt's (2014) construction of SL(2,C)-solutions of Hitchin's equations where the Higgs field is "simple." This is ongoing work.

L^q norms and nodal sets of Laplace eigenfunctions

Guillaume Roy-Fortin   [email] (Northwestern U, Mathematics)

Abstract: We will discuss a recent result that exhibits a relation between the average local growth of a Laplace eigenfunction on a compact, smooth Riemannian surface and the global size of its nodal (zero) set. More precisely, we provide a lower and an upper bound for the Hausdorff measure of the nodal set in terms of the average of the growth exponents of an eigenfunction on disks of small radius. Combined with Yau's conjecture and the work of Donnelly-Fefferman, the result implies that the average local growth of eigenfunctions on an analytic manifold with analytic metric is bounded by constants in the semi-classical limit.

Complex Polynomial Optimization

Cedric Josz (Univ. Paris VI)

Abstract: Finding the global optimum of a Hermitian symmetric polynomial on a compact semi-algebraic set is a non-deterministic polynomial-time hard problem. Thanks to D'Angelo's and Putinar's Positivstellensatz on odd-dimensional spheres, it breaks down to solving a sequence of complex semidefinite programming relaxations that grow tighter and tighter. This raises open questions regarding Hermitian sums of squares and the Quillen property of real algebraic varieties. We'll discuss these and look at numerical experiments on problems with several thousand complex variables. These consist in computing optimal power flows in the European high-voltage transmission network.

Moved to Thursday this week, see below

Radial Limits of Partial Theta and Similar Series

Kagan Kursungoz (Sabanci University, Turkey)

Abstract: We study unilateral series in a single variable q where its exponent is an unbounded increasing function, and the coefficients are periodic. Such series converge inside the unit disk. Quadratic polynomials in the exponent correspond to partial theta series. We compute limits of those series as the variable tends radially to a root of unity. The proofs use ideas from the q-integral and are elementary.

Some estimates for the parabolic Anderson model

Samy Tindel   [email] (Purdue University)

Abstract: We focus in this talk on stochastic heat equations whose noisy part is of the form u W, where u is the solution to the equation and W a rather general Gaussian noise. This model is usually called parabolic Anderson model, and is related to many physically relevant systems such as KPZ equation. We will first motivate our study, then show how to define and solve the stochastic heat equation. We shall derive some Feyman-Kac representations for the solution, either in a pathwise way of for moments. These Feyman-Kac formulae always involve some weighted Brownian local times, and our goal is to include a broad class of very irregular noises into the picture. Finally, we obtain some moment estimates which entail the so-called intermittency phenomenon. If time allows it, we shall also give some perspectives on future works in this direction, concerning a sharp description of the asymptotic spatial behavior of the solution.

On fake projective planes and related geometric problems.

Sai Kee Yeung (Purdue)

Abstract: The main purpose of the talk is to explain the recent classification of fake projective planes and some geometric applications. Fake projective plane was first introduced by David Mumford. It is defined to be a complex surface with the same Betti numbers as the complex projective plane, and has the smallest possible Euler number among all smooth surfaces of general type. Fake projective planes are classified by Gopal Prasad and myself, with the work of Donald Cartwright and Tim Steger. We would also mention some natural questions and applications, including existence of some exotic manifolds, question on existence of exceptional collection of objects on fake projective planes in a joint work with Ching-Jui Lai, and existence of surfaces of conjectured maximal canonical degree.

Expansion under minors

Kent Quanrud   [email] (CS Dept UIUC)

Abstract: In optimization, the edge density of a graph (i.e., the ratio of the number of edges to vertices) does not capture the computational intractability of graph problems. For example, planar graphs and constant degree expanders both have edge density bounded by a constant, but in combinatorial optimization the former is generally more tractable than the latter. A more refined approach is the notion of expansion under minors, developed recently by Nešetřil and Ossona de Mendez, which considers not only the edge density of a graph but the edge densities of its shallow minors. In this theory, planar graphs have small (constant) expansion under minors while expanders have large (exponential) expansion under minors. In this survey-style talk I will introduce the theory of expansion under minors and discuss some of its applications. The talk is motivated by recent work with Sariel Har-Peled in which we obtain polynomial time approximation schemes for problems such as minimum dominating set on classes of graphs with polynomial expansion (available at http://illinois.edu/~quanrud2/papers/low_density.pdf).