Mathematics

In this talk, which is based on Benjamin Weiss' Sofic groups and dynamical systems, I will give the definition of sofic groups which can be considered as common generalizations of amenable and residually finite groups, and discuss examples and some of their basic properties. Finally, we will give Weiss' alternative proof Gromov's theorem stating that sofic groups are surjunctive. This theorem settles Gottschalk's conjecture for sofic groups.

In the late 90's, Keating and Snaith used random matrix theory to predict the exact leading terms of conjectural asymptotic formulas for all integral moments of the Riemann zeta-function. Prior to their work, no number-theoretic argument or heuristic has led to such exact predictions for all integral moments. In 2015, Conrey and Keating revisited the approach of using divisor sum heuristics to predict asymptotic formulas for moments of zeta. Essentially, their method estimates moments of zeta using lower twisted moments. In this talk, I will discuss a rigorous random matrix theory analogue of the Conrey-Keating heuristic. This is ongoing joint work with Brian Conrey.

Historically, the notion of Pólya fields dates back to some works of George Pólya and Alexander Ostrowski, in 1919, on entire functions with integer values at integers; a number field $K$ with ring of integers $\mathcal{O}_K$ is called a Pólya field whenever the $\mathcal{O}_K$-module $\{f \in K[X] \, : \, f(\mathcal{O}_K) \subseteq \mathcal{O}_K \}$ admits an $\mathcal{O}_K$-basis with exactly one member from each degree. Pólya fields can be thought of as a generalization of number fields with class number one, and their classification of a specific degree has become recently an active research subject in algebraic number theory. In this talk, I will present some criteria for $K$ to be a Pólya field. Then I will give some results concerning Pólya fields of degrees $2$, $3$, and $6$.

Constructing and understanding the basic properties of Euclidean Yang-Mills theory is a fundamental problem in physics. It is also one of the Clay Institute's famous Millennium Prize problems in mathematics. The basic problem is not hard to understand. You can begin by describing a simple random function from a set of lattice edges to a group of matrices. Then you ask whether you can construct/understand a continuum analog of this object in one way or another. In addition to a truly enormous physics literature, this topic has inspired research within many major areas of mathematics: representation theory, random matrix theory, probability theory, differential geometry, stochastic partial differential equations, low-dimensional topology, graph theory and planar-map combinatorics.

Attempts to understand this problem in the 1970's and 1980's helped inspire the study of "random surfaces" including Liouville quantum gravity surfaces. Various relationships between Yang-Mills theory and random surface theory have been obtained over the years, but many of the most basic questions have remained out of reach. I will discuss our own recent work in this direction, as contained in two long recent papers relating "Wilson loop expectations" (the fundamental objects in Yang-Mills gauge theory) to "sums over spanning surfaces."

1. Wilson loop expectations as sums over surfaces on the plane (joint with Minjae Park, Joshua Pfeffer, Pu Yu)

2. Random surfaces and lattice Yang-Mills (joint with Sky Cao, Minjae Park)

The first paper explains how in 2D (where Yang-Mills theory is more tractable) one can interpret continuum Wilson loop expectations purely in terms of flat surfaces. The second explains a general-dimensional interpretation of the Wilson loop expectations in lattice Yang-Mills theory in terms of discrete-and not-necessarily-flat surfaces, a.k.a. embedded planar maps.

Speaker Biography:

Scott Sheffield is the Leighton Family Professor of Mathematics at MIT. He is a leading figure at the interface of mathematical physics and probability. He has held positions at Microsoft Research, Berkeley, the Institute for Advanced Study and New York University. He received the Rollo Davidson and Loève prizes in probability and has twice spoken at the International Congress of Mathematicians.

For every positive integer n, the quotient ring Z/nZ is the natural ring whose additive group is cyclic. The "multiplicative group modulo n" is the group of invertible elements of this ring, with the multiplication operation. As it turns out, many quantities of interest to number theorists can be interpreted as "statistics" of these multiplicative groups. For example, the cardinality of the multiplicative group modulo n is simply the Euler phi function of n; also, the number of terms in the invariant factor composition of this group is closely related to the number of primes dividing n. Many of these statistics have known distributions when the integer n is chosen at random (the Euler phi function has a singular cumulative distribution, while the Erdös–Kac theorem tells us that the number of prime divisors follows an asymptotically normal distribution). Therefore this family of groups provides a convenient excuse for examining several famous number theory results and open problems. We shall describe how we know, given the factorization of n, the exact structure of the multiplicative group modulo n, and go on to outline the connections to these classical statistical problems in multiplicative number theory.