Busenberg and Travis suggested in 1983 a population system that exhibits complete segregation of the species. This system can be rigorously derived from interacting particle systems in a mean-field-type limit. It consists of parabolic cross-diffusion equations with an indefinite diffusion matrix. It is known that this system can be formulated in terms of so-called entropy variables such that the transformed equations possess a positive semidefinite diffusion matrix. We consider in this talk the case of incomplete diffusion, which means that the diffusion matrix has zero eigenvalues, and the problem is not parabolic in the sense of Petrovskii. 

We show that the cross-diffusion equations can be written as a normal form of symmetric hyperbolic-parabolic type beyond the Kawashima-Shizuta theory. Using results for symmetric hyperbolic systems, we prove the existence of a unique local classical solution. As solutions may become discontinuous in finite time, only global solutions with very low regularity can be expected. We prove the existence of global dissipative measure-valued solutions satisfying a weak-strong uniqueness property. The proof is based on entropy methods and a finite-volume approximation with a mesh-dependent artificial diffusion. 

The seminar concerns the study of evolution equations on graphs, motivated by applications in data science and opinion dynamics. We will discuss graph analogues of the continuum nonlocal-interaction equation and interpret them as gradient flows with respect to a graph Wasserstein distance, using Benamou--Brenier formulation. The underlying geometry of the problem leads to a Finslerian gradient flow structure, rather than Riemannian, since the resulting distance on graphs is actually a quasi-metric. We will address the existence of suitably defined solutions, as well as their asymptotic behaviour when the number of vertices converges to infinity and the graph structure localises. The two limits lead to different dynamics. From a slightly different perspective, by means of a classical fixed-point argument, we can show the existence and uniqueness of solutions to a larger class of nonlocal continuity equations on graphs. In this context, we consider general interpolation functions of the mass on the edges, which give rise to a variety of different dynamics. Our analysis reveals structural differences with the more standard Euclidean space, as some analogous properties rely on the interpolation chosen. The latter study can be extended to equations on co-evolving graphs. The talk is based on works in collaboration with G. Heinze (Augsburg), L. Mikolas (Oxford), F. S. Patacchini (IFP Energies Nouvelles), A. Schlichting (University of Münster), and D. Slepcev (Carnegie Mellon University). 

The Nyström method offers an effective way to obtain low-rank approximation of SPD matrices, and has been recently extended and analyzed to nonsymmetric matrices (leading to the randomized, single-pass, streamable, cost-effective, and accurate alternative to the randomized SVD, and it facilitates the computation of several matrix low-rank factorizations. In this presentation, we take these advancements a step further by introducing a higher-order variant of Nyström's methodology tailored to approximating low-rank tensors in the Tucker format: the multilinear Nyström technique. We show that, by introducing appropriate small modifications in the formulation of the higher-order method, strong stability properties can be obtained. This algorithm retains the key attributes of the generalized Nyström method, positioning it as a viable substitute for the randomized higher-order SVD algorithm.

We propose a method to detect linear and nonlinear lead-lag relationships in stock returns.  Our approach uses pairwise Lévy-area and cross-correlation of returns to rank the assets from leaders to followers. We use the rankings to construct a portfolio that longs or shorts the followers based on the previous returns of the leaders, and the stocks are ranked every time the portfolio is rebalanced. The portfolio also takes an offsetting position on the SPY ETF so that the initial value of the portfolio is zero. Our data spans from 1963 to 2022 and we use an average of over 500 stocks to construct portfolios for each trading day. The annualized returns of our lead-lag portfolios are over  20%, and the returns outperform all lead-lag benchmarks in the literature. There is little overlap between the leaders and the followers we find and those that are reported in previous studies based on market capitalization, volume traded, and intra-industry relationships. Our findings support the slow information diffusion hypothesis; i.e., portfolios rebalanced once a day consistently outperform the bidiurnal, weekly, bi-weekly, tri-weekly, and monthly rebalanced portfolios.

If some structure, mathematical or otherwise, is giving you grief, then often the first thing to do is to attempt to break the offending object down into (finitely many) simpler pieces.

In group theory, when we speak of questions of *accessibility* we are referring to the ability to achieve precisely this. The idea of an 'accessible group' was first coined by Terry Wall in the 70s, and since then has left quite a mark on our field (and others). In this talk I will introduce the toolbox required to study accessibility, and walk you and your groups through some reasons to be accessible.

As part of the internal seminar schedule for Stochastic Analysis for this coming term, DPhil students have been invited to present on their works to date. Student talks are 20 minutes, which includes question and answer time.