Georgia Institute of Technology College of Sciences

We conclude the proof of the linearization theorem for fibered holomorphic maps by showing that the iteration scheme we proposed converges. If time allows, we will comment on related work by Mario Ponce and generalizations of the theorem for fibered holomorphic maps in higher dimensions.

Kronecker coefficients lie at the intersection of representation theory, algebraic combinatorics and, most recently, complexity theory. They count the multiplicities of irreducible representations in the tensor product of two other irreducible representations of the symmetric group. While their study was initiated almost 75 years, remarkably little is known about them. One of the major problems of algebraic combinatorics is to find an explicit positive combinatorial formula for these coefficients. Recently, this problem found a new meaning in the field of Geometric Complexity Theory, initiated by Mulmuley and Sohoni, where certain conjectures on the complexity of computing and deciding positivity of Kronecker coefficients are part of a program to prove the "P vs NP" problem. In this talk we will give an overview of this topic and we will describe several problems with some results on different aspects of the Kronecker coefficients. We will explore Saxl conjecture stating that the tensor square of certain irreducible representation of S_n contains every irreducible representation, and present a criterion for determining when a Kronecker coefficient is nonzero. In a more combinatorial direction, we will show how to prove certain unimodality results using Kronecker coefficients, including the classical Sylvester's theorem on the unimodality of q-binomial coefficients (as polynomials in q). We will also present some results on complexity in light of Mulmuley's conjectures. The presented results are based on joint work with Igor Pak and Ernesto Vallejo.

Abstract: (joint work with Peter Scholze) The proetale topology is a Grothendieck topology that is closely related to the etale topology, yet better suited for certain "infinite" constructions, typically encountered in l-adic cohomology. I will first explain the basic definitions, with ample motivation, and then discuss applications. In particular, we will see why locally constant sheaves in this topology yield a fundamental group that is rich enough to detect all l-adic local systems through its representation theory (which fails for the groups constructed in SGA on the simplest non-normal varieties, such as nodal curves).

Sources of single photons (as opposed to sources which produce on average a single photon) are of great current interest for quantum information processing. Perhaps surprisingly, it is not easy to produce a single photon efficiently and in a controlled way. Following earlier progress, recent experimental activity has resulted in the production of single photons by taking advantage of strong inter-particle interactions in cold atomic gases.I will show how the systematic use of the method of steepest descents can be used to understand the dynamics of the single photon source developed here at Georgia Tech and how this describes a kind of quantum scissors effect. In addition to the mathematical results, I will present the background quantum mechanics in a form suitable for a general audience. Joint work with Francesco Bariani and Paul Goldbart.