The goal of this talk will be to describe our recent result proving that the asymptotic dimension of the mapping class group of a closed surface is at most quadratic in the genus (building on and strengthening a prior result of Bestvina-Bromberg giving an exponential estimate). We obtain this result as a special case of a result about the asymptotic dimension of a general class of spaces, which we call hierarchically hyperbolic; this class includes hyperbolic spaces, mapping class groups, Teichmueller spaces endowed with either the Teichmuller or the Weil-Petersson metric, fundamental groups of non-geometric 3-manifolds, RAAGs, etc. We will discuss the general framework and a sketch of how this machinery provides new tools for studying special subclasses, such as mapping class groups. The results discussed are joint work with Mark Hagen and Alessandro Sisto.

# Asymptotic dimension of mapping class groups

# Fivebranes and 4-manifolds

I will talk about a novel invariant of four manifolds. Using the so called fivebrane of M-theory, we associate a two dimensional physical theory to a 4-manifold. From mathematical point of view, this theory can be thought of as a higher categorical data, which can be distilled to get more familiar quantities such as Donaldson-Witten invariants and Euler characteristics of instanton moduli spaces. This construction is useful from physics point of view as well, as the Kirby moves on 4-manifolds get mapped to non-trivial dualities of the 2d theory.

# Random walks on random graphs

We will discuss the behavior of the random walk on two random graph models: on one hand the random regular graph with fixed degree, and on the other hand the giant component of the supercritical Erdős-Renyi random graph with constant average degree. In the former case it is known that the walk mixes in logarithmic time and exhibits the cutoff phenomenon. In the latter case, while starting from the worst initial vertex---as shown by Fountoulakis-Reed and independently by Benjamini-Kozma-Wormald---both delays mixing and precludes cutoff, it turns out that starting from a fixed vertex induces the rapid mixing behavior of the regular case. For general locally-tree-like random graphs, we express the mixing time in terms of the speed and dimension of harmonic measure of random walk on the corresponding Galton-Watson tree.

Based on joint works with N. Berestycki, Y. Peres, and A. Sly

# Quantum Chromatic Numbers and the conjectures of Connes and Tsirelson

It is possible to characterize the chromatic number of a graph in terms of a game. It is the fewest number of colours for which a winning strategy exists using classical random variables to a certain graph colouring game. If one allows the players to use quantum experiments to generate their random outcomes, then for many graphs this game can be won with far fewer colours.

This leads to the definition of the quantum chromatic number of a graph. However, there are several mathematical models for the set of probability densities generated by quantum experiments and whether or not these models agree depends on deep conjectures of Connes and Tsirelson. Thus, there are potentially several "different" quantum chromatic numbers and computing them for various graphs gives us a combinatorial means to test these conjectures.In this talk I will present these ideas and some of the results in this area. I will only assume that the audience is familiar with the basics of Hilbert space theory and assume no background in quantum theory.

# Random loop representations of quantum spin systems and their universal behaviour 2-2

I will describe the representations of Tóth and Aizenman-Nachtergaele of quantum Heisenberg models. They constitute an example of “loop soup” models that display a universal behaviour in dimensions 3 and higher: At low temperatures, the system contains macroscopic loops and the joint distribution of their lengths is given by a Poisson-Dirichlet distribution. This can be understood (and calculated) by viewing the loop distribution as the invariant measure of an effective spit-merge process.

I will explain the relevant notions. If time permits, I will derive some consequences of this universal behaviour regarding the nature of symmetry breaking in quantum spin systems.

# Inference and Diagnostics for Respondent-Driven Sampling Data

Respondent-Driven Sampling is type of link-tracing network sampling used to study hard-to-reach populations. Beginning with a convenience sample, each person sampled is given 2-3 uniquely identified coupons to distribute to other members of the target population, making them eligible for enrollment in the study. This is effective at collecting large diverse samples from many populations.

Unfortunately, sampling is affected by many features of the network and sampling process, which complicate inference. In this talk, I highlight key methodological challenges arising from data collected in this manner. I then introduce key methods for diagnostics and inference in these settings, and describe new methods under development.

# The phase-diagram of the Blume-Capel-Haldane-Ising spin chain

We consider the one-dimensional spin chain for arbitrary spin $s$ on a periodic chain with $N$ sites, the generalization of the chain that was studied by Blume and Capel cite{bc}: $$H=sum_{i=1}^N left(a (S^z_i)^2+ b S^z_iS^z_{i+1} ight).$$ Although the Hamiltonian is trivially diagonal, it is actually not always obvious which eigenstate is the ground state. We show how to find the ground state for all regions of the parameter space and thus determine the phase diagram of the model. We observe the existence of solitons-like excitations and we show that the size of the solitons depends only on the ratio $a/b$ and not on the number of sites $N$ and not on boundary conditions.

# Numerical Computation on Curved Surfaces

Despite the appearance sometimes given in textbooks, not all differential equations are posed on straight lines and rectangles. This talk will introduce some easy-to-use techniques for computing numerical solutions to partial differential equations (PDEs) posed on curved surfaces and other general domains. I will show some applications in thin-film flows, reaction-diffusion equations, bulk-surface coupling, point clouds, and image processing. The talk will also outline how a close encounter with instability improved our understanding and numerical analysis of these methods.

# Random matrices, geometry and physics

This lecture is a journey amidst large random matrices, 3d Chern-Simons theory and 2d quantum gravity.

Eigenvalues of large random matrices provide a computable playground to investigate universal laws occuring in the statistics of a large number of strongly coupled, repelling variables. Their universal is such that they can be observed in various situations in physics and mathematics. I will describe general results about asymptotic analysis of a class of matrix models called "beta-ensembles", based on large deviation theory and functional analysis, and subsequent predictions about the Tracy-Widom law (i.e. the fluctuations of the maximum eigenvalue).

These general results can be applied to analyze the partition function of SU(N) Chern-Simons theory on simple 3-manifolds, namely quotients of S^3 by a finite group of isometries. The large N limit is described by a spectral curve which we can compute, and identify with the spectral curve of a relativistic Toda chain of type ADE. From the physics perspective, we propose a generalization of the Gopakumar-Vafa conjecture, i.e. a correspondence between Chern-Simons theory on S^3/Gamma, and topological strings on non-toric CY.

The all-order large N asymptotic expansion in these two problems are governed by a universal mathematical structure, called "topological recursion". I will give other examples of its applications, related to volumes of moduli spaces and to conformal field theory.

If time allows, I will describe a recent result about 2d quantum gravity, which shows agreement of a multifractal spectral describing the nesting statistics of self-avoiding loop configurations in "large random discrete surfaces" vs. "Conformal Loop Ensembles coupled to Liouville quantum gravity".

# Ph.D. Oral Defense - Mr. Jason Polak

The Department of Mathematics and Statistics invites you to attend the Ph.D. Oral Defense of Mr. Jason Polak.