Part II. A statistical physics analysis using random matrix theory and mean-field methods.

Mutations of quivers were introduced by Fomin and Zelevinsky in the context of cluster algebras. Since then, mutations appear (sometimes completely unexpectedly) in various domains of mathematics and physics. Using mutations of quivers, Barot and Marsh constructed a series of presentations of finite Coxeter groups as quotients of infinite Coxeter groups. I will discuss a geometric interpretation of this construction: these presentations give rise to a construction of geometric manifolds with large symmetry groups, in particular to some hyperbolic manifolds of relatively small volume with proper actions of Coxeter groups. If time permits, I will discuss a generalization of the construction of Barot and Marsh leading to a new invariant of bordered marked surfaces, and relation to extended affine Weyl groups. The talk is based on joint works with Anna Felikson, John Lawson and Michael Shapiro.

Given a graph G = (V,E), consider the set of all discrete-time, reversible Markov chains with equilibrium distribution uniform on V and transitions only across edges E of the graph. We establish a Cheeger-type inequality for the fastest mixing time using the vertex conductance of G. We also consider chains with almost-uniform invariant distribution. Time permitting, we also discuss a construction of a continuous-time chain with exactly-uniform invariant distribution and average jump-rate 1, and mixing time bounded by the d^2 log(n), where d is the graph diameter and n is the number of vertices.

I will discuss some of the recent developments in the Krylov complexity. In particular, I will focus on the applications of the Krylov basis techniques to the modular Hamiltonian evolution and I will discuss a new angle on entanglement entropy in QCD at high energies. Based on arXiv:2306.14732 [hep-th] and work in progress.

Part I, Introduction and theoretical basis: how time-reversing Langevin dynamics one can create images from white noise.

How often does a randomly chosen variety have a point? Answering this question depends on the family of varieties in question, how we decide to order them, and what kinds of points we are looking for. Motivated by rational points, we endeavor to explicitly describe how often a randomly chosen variety is everywhere locally soluble. When our family is described by the fibers of a suitable morphism, this likelihood is equal to the product of local probabilities at each place and in some cases may be computed exactly. In particular, in joint work with Lea Beneish we find that for almost 97% of integral binary sextic forms f(x,z), the superelliptic curve y^3 = f(x,z) is everywhere locally soluble, with the local factors described explicitly by rational functions. Time permitting, we will discuss ongoing work on determining how often a cubic hypersurface has a rational point.

Sequential Monte Carlo Samplers (SMCS) constitute a widely used class of SMC algorithms that calculate normalizing constants and simulate complex multi-modal target distributions. Typically, SMCS utilizes a process known as annealing, which propagates solutions from a tractable reference distribution to the intractable target through a continuous path of increasingly complex distributions. SMCS delivers state-of-the-art performance when adequately tuned, although this can pose a challenge for current tuning methods, yielding a random run-time and compromising the normalizing constant's unbiasedness.

In this talk, we intend to describe all the components of an SMCS algorithm and their influence on the variance of the normalizing constant. Specifically, we will demonstrate that SMCS exhibits fundamentally different behaviour in large particle and dense schedule limits. The dense schedule limit reveals the natural geometry induced by annealing, which can pinpoint optimal performance and tune the number of particles, number of annealing distributions, annealing schedules, the resampling schedule, and the path. Lastly, we propose an efficient, black-box algorithm for tuning SMCS that delivers optimal performance within a fixed, user-specified computation budget, all while preserving the unbiasedness of the normalizing constant.

We describe a new form of diagonalization for linear two point constant coefficient differential operators with arbitrary linear boundary conditions. Although the diagonalization is in a weaker sense than that usually employed to solve initial boundary value problems (IBVP), we show that it is sufficient to solve IBVP whose spatial parts are described by such operators. We argue that the method described may be viewed as a reimplementation of the Fokas transform method for linear evolution equations on the finite interval. The results are extended to multipoint and interface operators, including operators defined on networks of finite intervals, in which the coefficients of the differential operator may vary between subintervals, and arbitrary interface and boundary conditions may be imposed\DSEMIC differential operators with piecewise constant coefficients are thus included.

I will discuss a connection between harmonic analysis on hyperbolic n-manifolds and conformal field theory in n-1 dimensions. Used in one direction, this connection leads to new spectral bounds on hyperbolic manifolds. Used in the other direction, it offers a new viewpoint on the spectra data of conformal field theories.

There are many ways to engage with mathematics, and therefore many different forms of maths engagement aimed at different audiences. A brief survey will present a range of different practices and encourage thought about target audience. We'll consider how to design effective outreach, and how informal engagement with maths before university can influence the decision to study maths at university. From a practical point of view, I will describe a programme of maths engagement for secondary school students implemented in local schools and on campus.