The Wasserstein metric originates in the theory of optimal transport, and among many other applications, it provides a natural way to measure how evenly distributed a finite point set is. We give a survey of classical and more recent results that describe the behaviour of some random point processes in Wasserstein metric, including the eigenvalues of some random matrix models, and explain the connection to the logarithm of the characteristic polynomial of a random unitary matrix. We also discuss a simple random walk model on the unit circle defined in terms of a quadratic irrational number, which turns out to be related to surprisingly deep arithmetic properties of real quadratic fields.

The two-point Chowla conjecture predicts that $\sum_{x<n<2x} \lambda(n)\lambda(n+1) = o(x)$ as $x\to \infty$, where $\lambda$ is the Liouville function (a $\{\pm 1\}$-valued multiplicative function encoding the parity of the number of prime factors). While this remains an open problem, weaker versions of this conjecture are known. In this talk, we outline an approach initiated by Helfgott and Radziwill, which reformulates the problem in terms of bounding the eigenvalues of a certain matrix.

One curious little fact about the Riemann zeta function is that if you evaluate its derivatives at the zeros of zeta, then on average this is real and positive (even though the function is complex). This has been proven for some time now, but the aim of this talk is to generalise the question further (higher derivatives, complex moments) and gain insight using random matrix theory. The takeaway message will be that there are a multitude of different proof techniques in RMT, each with their own advantages

In this talk, I will introduce a dynamic model of a banking network in which the value of interbank obligations is continuously adjusted to reflect counterparty default risk. An interesting feature of the model is that the credit value adjustments increase volatility during downturns, leading to endogenous distress contagion. The counterparty default risk can be computed backwards in time from the obligations' maturity date, leading to a specification of the model in terms of a forward-backward stochastic differential equation (FBSDE), coupled through the banks' default times. The singular nature of this coupling, makes a probabilistic analysis of the FBSDE challenging. So, instead, we derive a characterisation of the default probabilities through a cascade of partial differential equations (PDE). Each PDE represents a configuration with a different number of defaulted banks and has a free boundary that coincides with the banks' default thresholds. We establish classical well-posedness of this PDE cascade, from which we derive existence and uniqueness of the FBSDE.