Georgia Institute of Technology College of Sciences

Stability is a multivariate generalization for real-rootedness in univariate polynomials. Within the past ten years, the theory of stable polynomials has contributed to breakthroughs in combinatorics, convex optimization, and operator theory. I will introduce a generalization of stability, called complete log-concavity, that satisfies many of the same desirable properties. These polynomials were inspired by work of Adiprasito, Huh, and Katz on combinatorial Hodge theory, but can be defined and understood in elementary terms. The structure of these polynomials is closely tied with notions of discrete convexity, including matroids, submodular functions, and generalized permutohedra. I will discuss the beautiful real and combinatorial geometry underlying these polynomials and applications to matroid theory, including a proof of Mason’s conjecture on numbers of independent sets. This is based on joint work with Nima Anari, Kuikui Liu, and Shayan Oveis Gharan.

(*Refreshments available at 2:30pm before the colloquium.*)

Milnor K-theory is a field invariant that originated as an attempt to study algebraic K-theory. Instead, Milnor K-theory has proved to have many other applications, including Galois cohomology computations, Voevodsky's proof of the Bloch-Kato conjecture, and Kato's higher class field theory. In this talk, we will go over the basic definitions and theorems of Milnor K-theory. We will also discuss some of these applications.

In a fractional coloring, vertices of a graph are assigned subsets of the [0, 1]-interval such that adjacent vertices receive disjoint subsets. The fractional chromatic number of a graph is at most k if it admits a fractional coloring in which the amount of "color" assigned to each vertex is at least 1/k. We investigate fractional colorings where vertices "demand" different amounts of color, determined by local parameters such as the degree of a vertex. Many well-known results concerning the fractional chromatic number and independence number have natural generalizations in this new paradigm. We discuss several such results as well as open problems. In particular, we will sketch a proof of a "local demands" version of Brooks' Theorem that considerably generalizes the Caro-Wei Theorem and implies new bounds on the independence number. Joint work with Luke Postle.

Casson invariant is defined for the class of oriented integral homology 3-spheres. It satisfies certain properties, and reduce to Rohlin invariant after mod 2. We will define Casson invariant as half of the algebraic intersection number of irreducible representation spaces (space consists of representations of fundamental group to SU(2)), and then prove this definition satisfies the expected properties.