# Mathematics Colloquium

The Mathematics Department Colloquium is a quarterly series of invited speakers from all mathematical fields and geared toward a broad audience. Talks are generally every Tuesday from 4-5 PM, with an informal tea beforehand. For the duration of COVID-19, it will be held via Zoom, with a meet-and-greet ~15m before the talk-- bring your own beverage.

**Fall 2021 Math Colloquium**

**THURSDAYS @ 4pm California Time**

**"tea" (meet-and-greet) @ ~15m before the talk**

**Meeting ID: 898 285 3979 Passcode: 107881 [****Direct link****]**

**Schedule ***(click dates for title and abstract)*

**Yuliy Baryshnikov, University of Illinois at Urbana-Champaign**

*Based on joint work with Robin Pemantle and Steve Melczer*

*Frozen regions are a prominent feature in many asymptotic problems of "integrable probability", a class of systems exhibiting spatial phase transitions between different regimes. (An archetypal example is the Aztec diamond dimer tiling model.)*

**Frozen regions and hyperbolic polynomials**

In the (frequent) situation when the multivariate generating functions for the order parameter is known and is rational, the description of the phases and boundaries between them reduces to the well studied techniques (by Petrovsky, Leray,..., Atiyah-Bott-Garding) of inverse Fourier transforms of rational functions with homogeneous hyperbolic polynomials.

I will outline the basic tools of the trade ab ovo, ending, time permitting, with the recent appearances of lacunae on the scene.

**Daniel Erman, University of Wisconsin**

*Limits of polynomials rings*

I will discuss the limit of a polynomial ring in n variables as n goes to infinity, and how this can be used to study complexity problems in algebra and algebraic geometry.

**Lisa Carbone, Rutgers University**

*A Lie group analog for the monster Lie algebra*

The Monster Lie algebra m is an infinite dimensional Lie algebra constructed by Borcherds as part of his program to solve the Conway-Norton Monstrous Moonshine Conjecture. We describe how one may approach the problem of associating a Lie group analog for m and we outline some constructions. This is joint work with Abid Ali, Elizabeth Jurisich and Scott H. Murray.

**Pierre Albin, University of Illinois at Urbana-Champaign**

*The signature of stratified spaces*

**Stefano Profumo, UCSC (Physics) and SCIPP (Santa Cruz Institute for Particle Physics)**

*What is the Dark Matter?*

Four fifths of the matter in the universe is made of something completely different from the "ordinary matter" we know and love. I will explain why this "dark matter" is an unavoidable ingredient to explain the universe as we observe it, and I will describe what the fundamental, particle nature of the dark matter could possibly consist of. I will then give an overview of strategies to search for dark matter as a particle, describe a few examples of possible hints of discovery, and outline ways forward in this exciting hunt.

**William Minicozzi, MIT**

*The diffeomorphism group of non-compact manifolds and singularities in Ricci flow*

Comparing and recognizing metrics can be extraordinarily difficult because of the group of diffeomorphisms. Two metrics, that could even be the same, could look completely different in different coordinates. For compact manifolds, various techniques exist for understanding this. However, for non-compact manifolds, no general techniques exist; controlling growth is a major obstruction. We introduce a new way of dealing with the gauge group of non-compact spaces by solving a nonlinear system of PDEs and proving optimal bounds for solutions. The PDE produces a diffeomorphism that fixes an appropriate gauge in the spirit of the slice theorem for group actions. The construction relies on sharp polynomial growth bounds for a linear system operator. We use this to solve a well known open problem in Ricci flow. This is joint work with Toby Colding.

**Richard Ng, Louisiana State University**

*Arithmetic of Modular Tensor Categories*

Modular categories arise naturally in rational conformal field theory (RCFT), quantum invariants of knots and 3-manifolds, and quantum computing. Mathematically, they are categorical generalizations of nondegenerate quadratic forms on finite abelian groups. The S and T-matrices of a modular category are a reincarnation of the Weil representation of a quadratic form. The celebrated Verlinde formula for the S-matrix and Vafa’s theorem on the finiteness of the order of the T-matrix are striking arithmetic properties of modular categories. In this talk, we will give a general introduction to modular categories, their associated representations of SL(2,Z), some recent results, and their applications to RCFT and quantum computing.