We will present some of the remarkable properties of eigenvalue trajectories for rank-one perturbations of random matrices, with an emphasis on two models of particular interest, namely weakly non-Hermitian and weakly non-unitary matrices. In both cases, precise estimates can be obtained for the critical timescale at which an outlier can be observed with high probability. We will outline the proofs of these results and highlight their significance in connection with quantum chaotic scattering. (Based on joint works with L. Erdös and J. Reker)

Multicomponent flows, i.e. mixtures, can be modeled effectively using the Onsager-Stefan-Maxwell (OSM) equations. The OSM equations can account for a wide variety of phenomena such as diffusive, convective, non-ideal mixing, thermal, pressure and electrochemical effects for steady and transient multicomponent flows. I will first introduce the general OSM framework and a finite element discretisation for multicomponent diffusion of ideal gasses. Then I will show two ways of preconditioning the multicomponent diffusion problem for various boundary conditions. Time permitting, I will also discuss how this can be extended to the non-ideal, thermal, and nonisobaric settings.

Tensor methods are methods for unconstrained continuous optimization that can incorporate derivative information of up to order p > 2 by computing a step based on the pth-order Taylor expansion at each iteration. The most important among them are regularization-based tensor methods which have been shown to have optimal worst-case iteration complexity of finding an approximate minimizer. Moreover, as one might expect, this worst-case complexity improves as p increases, highlighting the potential advantage of tensor methods. Still, the global complexity results only guarantee pessimistic sublinear rates, so it is natural to ask how local rates depend on the order of the Taylor expansion p. In the case of strongly convex functions and a fixed regularization parameter, the answer is given in a paper by Doikov and Nesterov from 2022: we get pth-order local convergence of function values and gradient norms. 
The value of the regularization parameter in their analysis depends on the Lipschitz constant of the pth derivative. Since this constant is not usually known in advance, adaptive regularization methods are more practical. We extend the local convergence results to locally strongly convex functions and fully adaptive methods. 
We discuss how for p > 2 it becomes crucial to select the "right" minimizer of the regularized local model in each iteration to ensure all iterations are eventually successful. Counterexamples show that in particular the global minimizer of the subproblem is not suitable in general. If the right minimizer is used, the pth-order local convergence is preserved, otherwise the rate stays superlinear but with an exponent arbitrarily close to one depending on the algorithm parameters.

The Parker conjecture, which explores whether magnetic fields in perfectly conducting plasmas can develop tangential discontinuities during magnetic relaxation, remains an open question in astrophysics. Helicity conservation provides a topological barrier against topologically nontrivial initial data relaxing to a trivial solution. Preserving this mechanism is therefore crucial for numerical simulation.  

This paper presents an energy- and helicity-preserving finite element discretization for the magneto-frictional system for investigating the Parker conjecture. The algorithm enjoys a discrete version of the topological mechanism and a discrete Arnold inequality. 
We will also discuss extensions to domains with nontrivial topology.

This is joint work with Prof Patrick Farrell, Dr Kaibo Hu, and Boris Andrews

In Bayesian inverse problems, it is common to consider several hyperparameters that define the prior and the noise model that must be estimated from the data. In particular, we are interested in linear inverse problems with additive Gaussian noise and Gaussian priors defined using Matern covariance models. In this case, we estimate the hyperparameters using the maximum a posteriori (MAP) estimate of the marginalized posterior distribution. 

However, this is a computationally intensive task since it involves computing log determinants.  To address this challenge, we consider a stochastic average approximation (SAA) of the objective function and use the preconditioned Lanczos method to compute efficient function evaluation approximations. 

We can therefore compute the MAP estimate of the hyperparameters efficiently by building a preconditioner which can be updated cheaply for new values of the hyperparameters; and by leveraging numerical linear algebra tools to reuse information efficiently for computing approximations of the gradient evaluations.  We demonstrate the performance of our approach on inverse problems from tomography. 

In recent years, some progress has been made in the development of finite element complexes, particularly in the discretization of BGG complexes in two and three dimensions, including Hessian complexes, elasticity complexes, and divdiv complexes. In this talk, I will discuss distributional complexes in two and three dimensions. These complexes are simply constructed using geometric concepts such as vertices, edges, and faces, and they share the same cohomology as the complexes at the continuous level, which reflects that the discretization is structure preserving. The results can be regarded as a tensor generalization of the Whitney forms of the finite element exterior calculus. This talk is based on joint work with Snorre Christiansen (Oslo), Kaibo Hu (Edinburgh), and Qian Zhang (Michigan).

On arbitrary Carnot groups, the only hypoelliptic Hodge-Laplacians on forms that have been introduced are 0-order pseudodifferential operators constructed using the Rumin complex.  However, to address questions where one needs sharp estimates, this 0-order operator is not suitable. Indeed, this is a rather difficult problem to tackle in full generality, the main issue being that the Rumin exterior differential is not homogeneous on arbitrary Carnot groups. In this talk, I will focus on the specific example of the free Carnot group of step 3 with 2 generators, where it is possible to introduce different hypoelliptic Hodge-Laplacians on forms. Such Laplacians can be used to obtain sharp div-curl type inequalities akin to those considered by Bourgain & Brezis and Lanzani & Stein for the de Rham complex, or their subelliptic counterparts obtained by Baldi, Franchi & Pansu for the Rumin complex on Heisenberg groups

Von Neumann algebras which are not matrix algebras, yet still possess a unique trace, form a basic class called II$_1$ factors. The set of asymptotically commuting elements (or, the relative commutant of the algebra within its own ultrapower), dubbed the central sequence algebra, can take many different forms. In this talk, we discuss an elementary class of II$_1$ factors whose central sequence algebra is again a II$_1$ factor. We show that the class of infinitely generic II$_1$ factors possess this property, and ask some related questions about properties of other existentially closed II$_1$ factors. This is based on joint work with Isaac Goldbring, David Jekel, and Srivatsav Kunnawalkam Elayavalli.