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.
 

We consider a system of nonlinear partial differential equations modelling the steady motion of an incompressible non-Newtonian fluid, which is chemically reacting. The governing system consists of a steady convection-diffusion equation for the concentration and the generalized steady Navier–Stokes equations, where the viscosity coefficient is a power-law type function of the shear-rate, and the coupling between the equations results from the concentration-dependence of the power-law index. This system of nonlinear partial differential equations arises in mathematical models of the synovial fluid found in the cavities of moving joints. We construct a finite element approximation of the model and perform the mathematical analysis of the numerical method. Key technical tools include discrete counterparts of the Bogovski operator, De Giorgi’s regularity theorem and the Acerbi–Fusco Lipschitz truncation of Sobolev functions, in function spaces with variable integrability exponents.

In this talk we discuss the convergence rate of the Newton method for finding the singularity point on vetor fields. It is well-known that the Newton Method has local quadratic convergence rate with nonsingularity and Lipschitz condition. Here we release Lipschitz condition. With only nonsingularity, the Newton Method has superlinear convergence. If we have enough time, we can quickly give the damped Newton method on finding singularity on vector fields with superlinear convergence under nonsingularity condition only.

In this talk we consider the issue of mass loss in fragmentation models due to 'shattering'. As a solution we propose a hybrid discrete/continuous model whereby the smaller particles are considered as having discrete mass, whilst above a certain cut-off, mass is taken to be a continuous variable. The talk covers the development of such a model, its initial analysis via the theory of operator semigroups and its numerical approximation using a finite volume discretisation.

We explore the use of applying multiple preconditioners for solving linear systems arising in simulations of incompressible two-phase flow. In particular, we use a selective MPGMRES algorithm, for which the search space grows linearly throughout the iterative solver, and block preconditioners based on Schur complement approximations

We introduce a discontinuous Galerkin finite element method (DGFEM) for Hamilton–Jacobi–Bellman equations on piecewise curved domains, and prove that the method is consistent, stable, and produces optimal convergence rates. Upon utilising a long standing result due to N. Krylov, we may characterise the Monge–Ampère equation as a HJB equation; in two dimensions, this HJB equation can be characterised further as uniformly elliptic HJB equation, allowing for the application of the DGFEM

We propose a new parameter estimation technique for SDEs, based on the inverse problem of finding a forward operator describing the evolution of temporal data. Nonlinear dynamical systems on a state-space can be lifted to linear dynamical systems on spaces of higher, often infinite, dimension. Recently, much work has gone into approximating these higher-dimensional systems with linear operators calculated from data, using what is called Dynamic Mode Decomposition (DMD). For SDEs, this linear system is given by a second-order differential operator, which we can quickly calculate and compare to the DMD operator.

In 2016-17, Fjordholm, Kappeli, Mishra and Tadmor developed a numerical method by which one could compute measure-valued solutions to systems of hyperbolic conservation laws with either measure-valued or deterministic initial data. In this talk I will discuss the ideas behind this method, and discuss how it can be adapted to systems of quasi-linear parabolic PDEs whose nonlinearity fails to satisfy a monotonicity condition.

The basic mathematical models that describe the behavior of fluid flows date back to the eighteenth century, and yet many phenomena observed in experiments are far from being well understood from a theoretical viewpoint. For instance, especially challenging is the study of fundamental stability mechanisms when weak dissipative forces (generated, for example, by molecular friction) interact with advection processes, such as mixing and stirring. The goal of this talk is to have an overview on recent results on a variety of aspects related to hydrodynamic stability, such as the stability of vortices and laminar flows, the enhancement of dissipative force via mixing, and the statistical description of turbulent flows.

Serre's uniformity question concerns the possible ways the Galois group of Q can act on the p-torsion of an elliptic curve over Q. In this talk I will survey what is known about this question, and describe two recent results related to the Chabauty-Kim method. The first, which is joint work with Jennifer Balakrishnan, Steffen Muller, Jan Tuitman and Jan Vonk, completes the classification of elliptic curves over Q with split Cartan level structure. The second, which is work in progress with Samuel Le Fourn, Samir Siksek and Jan Vonk, concerns the applicability of the Chabauty-Kim method in determining the elliptic curves with non-split Cartan level structure.