Oxford University

Low-rank plus sparse matrices arise in many data-oriented applications, most notably in a foreground-background separation from a moving camera. It is known that low-rank matrix recovery from a few entries (low-rank matrix completion) requires low coherence (Candes et al 2009) as in the extreme cases when the low-rank matrix is also sparse, where matrix completion can miss information and be unrecoverable. However, the requirement of low coherence does not suffice in the low-rank plus sparse model, as the set of low-rank plus sparse matrices is not closed. We will discuss the relation of non-closedness of the low-rank plus sparse model to the notion of matrix rigidity function in complexity theory.

A simple experiment in the field of electrochemistry involves  controlling the applied potential in an electrochemical cell. This  causes electron transfer to take place at the electrode surface and in turn this causes a current to flow. The current depends on parameters in  the system and the inverse problem requires us to estimate these  parameters given an experimental trace of the current. We briefly  describe recent work in this area from simple least squares approximation of the parameters, through bootstrapping to estimate the distributions of the parameters, to MCMC methods which allow us to see correlations between parameters.

The question of what happens to the eigenvalues of a matrix after an additive perturbation has a long history, with notable contributions from Wilkinson, Sorensen, Golub, H\"ormander, Ipsen and Mehrmann, among many others. If the perturbed matrix $C \in \mathbb{C}^{n \times n}$ is given by $C = A + B$, these theorems typically consider the case where $A$ and/or $B$ are symmetric and $B$ has rank one. In this talk we will prove a theorem that bounds the number of distinct eigenvalues of $C$ in terms of the number of distinct eigenvalues of $A$, the diagonalisability of $A$, and the rank of $B$. This new theorem is more general in that it applies to arbitrary matrices $A$ perturbed by matrices of arbitrary rank $B$. We will also discuss various refinements of my bound recently developed by other authors.
 

I will explain a beautiful link between differential and algebraic geometry, called the Kempf-Ness Theorem, which says that the natural notions of "quotient spaces" in the symplectic and algebraic categories can often be identified. The result will be presented in its most general form where actions are not necessarily free and hence I will also introduce the notion of stratified spaces.

Many problems arising in Physics can be posed as minimisation of energy functionals under linear partial differential constraints. For example, a prototypical example in the Calculus of Variations is given by functionals defined on curl-free fields, i.e., gradients. Most work done subject to more general constraints met significant difficulty due to the lack of associated potentials. We show that under the constant rank assumption, which holds true of almost all examples of constraints investigated in connection with lower-semicontinuity, linear constraints admit a potential in frequency space. As a consequence, the notion of A-quasiconvexity, which involves testing with periodic fields leading to difficulties in establishing sufficiency for weak sequential lower semi-continuity, can be tested against compactly supported fields. We will indicate how this can simplify the general framework.

We will discuss nonlinear cross-diffusion models describing cell motility of two distinct populations. The continuum PDE model is derived systematically from a stochastic discrete model consisting of impenetrable diffusing spheres. In this talk, I will outline the derivation of the cross-diffusion model, discuss some of its features such as the gradient-flow structure, and show numerical results comparing the discrete stochastic system to the derived model.

We describe a locally finite graph naturally associated to each knot type K, called the Reidemeister graph. We determine several local and global properties of this graph and prove that the graph-isomorphism type is a complete knot invariant up to mirroring. Lastly (time permitting), we introduce another object, relating the Reidemeister and Gordian graphs, and briefly present an application to the study of DNA.

Understanding the distribution of subgraph counts has long been a central question in the study of random graphs. In this talk, we consider the distribution of Sn, the number of K4 subgraphs, in the Erdös Rényi random graph G(n, p). When the edge probability p \in (0, 1) is constant, a classical central limit theorem for Sn states that (Sn−µn)/σn converges in distribution. We establish a stronger form of convergence, namely the corresponding local limit theorem, which is joint work with O. Riordan.
 

In this joint work with D. Ciubotaru, we introduce the notion of local and global indices of Dirac operators for a rational Cherednik algebra H, with underlying reflection group G. In the local theory, I will report on some relations between the (local) Dirac index of a simple module in category O, the graded G-character and the composition series polynomials for standard modules. In the global theory, we introduce an "integral-reflection" module over which we define and compute the index of a (global) Dirac operator and show that the index is independent of the parameters. If time permits, I will discuss some local-global relations.