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.

Perturbative expansions in quantum theory, particularly in quantum field theory and string theory, are typically factorially divergent due to underlying non-perturbative sectors. Resurgence provides a universal toolbox to access the non-perturbative effects hidden within the perturbative series, producing a collection of exponentially small corrections. Under special assumptions, the non-perturbative data extracted via resurgent methods exhibit intrinsic number-theoretic structures that are deeply rooted in the symmetries of the theory. The framework of modular resurgence aims to formalise this observation. In this talk, I will first introduce the systematic, algebraic approach of resurgence to the problem of divergences and describe the emerging bridge between the resurgence of q-series and the analytic and number-theoretic properties of L-functions and quantum modular forms. I will then apply it to the spectral theory of quantum operators associated with toric Calabi-Yau threefolds. Here, a complete realisation of the modular resurgence paradigm is found in the study of the spectral trace of local P^2, where the asymptotics at weak and strong coupling are captured by certain q-series, and is generalised to all local weighted projective planes. This talk is based on arXiv:2212.10606, 2404.10695, 2404.11550, and work to appear soon.

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.

Multiparameter persistent homology is a generalization of persistent homology that allows for more than a single filtration function. Such constructions arise naturally when considering data with outliers or variations in density, time-varying data, or functional data. Even though its algebraic roots are substantially more complicated, several new invariants have been proposed recently. In this talk, I will go over such invariants, as well as their stability, vectorizations and implementations in statistical machine learning.