TATERS

 Strength is a fundamental invariant measuring non-degeneracy for polynomials in many variables. First applied to asymptotically counting integer solutions of polynomial equations via the Hardy-Littlewood circle method, strength has more recently been used in additive combinatorics for studying distribution of polynomials over finite fields. We prove that, over many fields, the strength of a polynomial is equivalent to the codimension of its singular locus. This substantially simplifies the proofs of various known results and partially resolves a conjecture of Adiprasito, Kazhdan and Ziegler. No prior knowledge of the Hardy-Littlewood circle method will be assumed. Based on joint work with Benjamin Baily.

The Laplace expansion expresses the nxn determinant as a sum of n products. What is the smallest possible number of products? We study two natural interpretations of this question, and obtain nontrivial results.

As a corollary, we obtain the first asymptotic separation between the partition rank and analytic rank of d-tensors, as d grows.

Mirroring Herb Wilf's much-cherished and still-wide-open question which polynomials are chromatic polynomials?, we give a brief survey of both the classification problem (the afore-mentioned question) and the detection problem (does a chromatic polynomial determine the graph?) for chromatic polynomials, which enumerate proper colorings of a given graph in terms of the number of colors. Inspired in part by the failure of the detection problem, Stanley introduced in the 1990s chromatic symmetric functions, which in turn opened up various lines of research. We introduce and study a q-version of the chromatic polynomial of a given graph G, defined as the sum of q^{l*c(v)} where l is a fixed integral linear form and the sum is over all proper n-colorings c of G. This turns out to be a polynomial in the q-integer [n]_q, with coefficients that are rational functions in q, and can be seen as an amphibian between chromatic polynomials and chromatic symmetric functions. We will exhibit several other structural results for q-chromatic polynomials and offer a strengthened version of Stanley's conjecture that the chromatic symmetric function distinguishes trees.

This talk is based on joint projects with Esme Bajo and Andrés Vindas-Meléndez.

In modern physics, all physical interactions propagate through fields permeating spacetime.  Even before quantizing these fields, the classical theory of these fields has a rich mathematical structure.  Two key initial observations are: (1) the dynamics of these fields is governed by interesting partial differential equations; and (2) these equations exhibit symmetries which do not come from the symmetries of spacetime.  Gauge theory is the study of fields which exhibit such symmetries. (The term “gauge” in physics is just a choice of trivialization of some bundle.)

In this talk, I will give a tour of the history of gauge theory, a subbranch of differential geometry culminating in a breakthrough theorem in 4-manifold topology in 1983.  The first gauge theory was Maxwell’s theory of electromagnetism. The physical observables of this theory are the electric and magnetic fields, described in terms of auxiliary electric potentials and magnetic potentials which are required to satisfy a hyperbolic partial differential equation (in a vacuum, this PDE is simply the wave equation). We will discuss Maxwell’s equations, the Yang-Mills equations, the anti-self-dual Yang-Mills equations, culminating in Donaldson’s beautiful gauge-theoretic proof that not all topological 4-manifolds have a smooth structure.  

I will conclude by briefly talking about my research on the asymptotic geometry of the Hitchin moduli space.

Prerequisites: One of the joys and challenges of gauge theory is its interdisciplinary nature.  Though I will introduce it, prior exposure to manifolds and vector bundles would certainly be helpful.

The surface skein module of a 3-manifold M was first defined by Asaeda–Frohman and Kaiser. This is spanned by embedded decorated surfaces in M which are considered up to local relations. The decorations and relations are determined by a commutative Frobenius algebra. This admits a description as a topological quantum field theory (TQFT) which assigns surface skein modules to 3-manifolds. Using an inductive construction following Walker, the TQFT can be extended partially to dimension 4.

In this talk, I will explain the basics of surface skein theory, and sketch the construction of the extension to invariants of 4-manifolds with a given handle decomposition. The talk will be guided by concrete examples.

A family F of compact n-manifolds is locally combinatorially defined (LCD) if there is a finite number of triangulated n-balls such that every manifold in F has a triangulation that locally looks like one of these n-balls. In joint work with Daryl Cooper and Priyam Patel, we show that LCD is equivalent to the existence of a compact branched n-manifold W, such that F is precisely those manifolds that immerse into W. In this way, W can be thought of as a universal branched manifold for F. In current and future work, we use this equivalence to show that, for each of the eight Thurston geometries, the family of closed 3-manifolds admitting that geometry is LCD. In this talk, I will present the main ideas of the proof of the equivalence and if time permits, construct branched 3-manifolds for a few of the geometries.

Fuchsian groups arise naturally as groups of covering transformations of hyperbolic surfaces. One may view them as images of discrete faithful representation of free groups and surface groups into PSL(2,R). The study of hyperbolic surfaces and deformation spaces of Fuchsian groups is a rich and classical subject. One may naturally generalize this to the study of groups of covering transformations of hyperbolic manifolds and this also has a beautiful, well-developed theory especially in dimension three.

Introduced in 2006, the theory of Anosov representations into semi-simple Lie groups (e.g. PSL(d,R)) has emerged as a higher rank analogue of the theory of Fuchsian groups. Our talk will begin by recalling some of the classical facts about Fuchsian groups. We will then gently introduce the subject of Anosov representations and their emerging theory.