University of Sussex

In this talk, I review some recent results obtained for black holes using
effective field theory methods applied to quantum gravity, in particular the
unique effective action. Black holes are complex thermodynamical objects
that not only have a temperature but also have a pressure. Furthermore, they
have quantum hair which provides a solution to the black hole information
paradox.

We study expectations of powers and correlations for characteristic polynomials of N x N non-Hermitian random matrices. This problem is related to the analysis of planar models (log-gases) where a Gaussian (or other) background measure is perturbed by a finite number of point charges in the plane. I will discuss the critical asymptotics, for example when a point charge collides with the boundary of the support, or when two point charges collide with each other (coalesce) in the bulk. In many of these situations, we are able to express the results in terms of Painlevé transcendents. The application to certain d-fold rotationally invariant models will be discussed. This is joint work with Alfredo Deaño (University of Kent).

We analyze a fully discrete numerical scheme for solving a parabolic PDE on a moving surface. The method is based on a diffuse interface approach that involves a level set description of the moving surface. Under suitable conditions on the spatial grid size, the time step and the interface width we obtain stability and error bounds with respect to natural norms. Test calculations are presented that confirm our analysis.

In this talk, I will present a novel solution strategy to efficiently and accurately compute approximate solutions to semilinear optimal control problems, focusing on the optimal control of phase field formulations of geometric evolution laws.
The optimal control of geometric evolution laws arises in a number of applications in fields including material science, image processing, tumour growth and cell motility.
Despite this, many open problems remain in the analysis and approximation of such problems.
In the current work we focus on a phase field formulation of the optimal control problem, hence exploiting the well developed mathematical theory for the optimal control of semilinear parabolic partial differential equations.
Approximation of the resulting optimal control problem is computationally challenging, requiring massive amounts of computational time and memory storage.
The main focus of this work is to propose, derive, implement and test an efficient solution method for such problems. The solver for the discretised partial differential equations is based upon a geometric multigrid method incorporating advanced techniques to deal with the nonlinearities in the problem and utilising adaptive mesh refinement.
An in-house two-grid solution strategy for the forward and adjoint problems, that significantly reduces memory requirements and CPU time, is proposed and investigated computationally.
Furthermore, parallelisation as well as an adaptive-step gradient update for the control are employed to further improve efficiency.
Along with a detailed description of our proposed solution method together with its implementation we present a number of computational results that demonstrate and evaluate our algorithms with respect to accuracy and efficiency.
A highlight of the present work is simulation results on the optimal control of phase field formulations of geometric evolution laws in 3-D which would be computationally infeasible without the solution strategies proposed in the present work.

Linear elasticity can be rigorously derived from finite elasticity in the case of small loadings in terms of \Gamma-convergence. This was first done by Dal Maso-Negri-Percivale in the case of one-well energies with super-quadratic growth. This has been later generalised to different settings, in particular to the case of multi-well energies where the distance between the wells is very small (comparable to the size of the load). I will discuss recent developments in the case when the distance between the wells is arbitrary. In this context linear elasticity can be derived by adding to the multi-well energy a singular higher order term which penalises jumps from one well to another. The size of the singular term has to satisfy certain scaling assumptions which turn out to be optimal. (This is joint work with Alicandro, Dal Maso and Lazzaroni.) 

In this seminar I will present a semi-langrangian discretisation of the Monge-Ampère operator, which is of interest in optimal transport 
and differential geometry as well as in related fields of application.

I will discuss the proof of convergence to viscosity solutions. To address the challenge of uniqueness and convexity we draw upon the classical relationship with Hamilton-Jacobi-Bellman equations, which we extend to the viscosity setting. I will explain that the monotonicity of semi-langrangian schemes implies that they possess large stencils, which in turn requires careful treatment of the boundary conditions.

The contents of the seminar is based on current work with X Feng from the University of Tennessee.

I will present some recent results concerning the higher gradient integrability of

σ-harmonic functions u with discontinuous coefficients σ, i.e. weak solutions of

div(σ∇u) = 0. When σ is assumed to be symmetric, then the optimal integrability

exponent of the gradient field is known thanks to the work of Astala and Leonetti

& Nesi. I will discuss the case when only the ellipticity is fixed and σ is otherwise

unconstrained and show that the optimal exponent is attained on the class of

two-phase conductivities σ: Ω⊂R27→ {σ1,σ2} ⊂M2×2. The optimal exponent

is established, in the strongest possible way of the existence of so-called

exact solutions, via the exhibition of optimal microgeometries.

(Joint work with V. Nesi and M. Ponsiglione.)

Cell migration is of vital importance in many biological studies, hence robust cell tracking algorithms are needed for inference of dynamic features from (static) in vivo and in vitro experimental imaging data of cells migrating. In recent years much attention has been focused on the modelling of cell motility from physical principles and the development of state-of-the art numerical methods for the simulation of the model equations. Despite this, the vast majority of cell tracking algorithms proposed to date focus solely on the imaging data itself and do not attempt to incorporate any physical knowledge on cell migration into the tracking procedure. In this study, we present a mathematical approach for cell tracking, in which we formulate the cell tracking problem as an inverse problem for fitting a mathematical model for cell motility to experimental imaging data. The novelty of this approach is that the physics underlying the model for cell migration is encoded in the tracking algorithm. To illustrate this we focus on an example of Zebrafish (Danio rerio's larvae} Neutrophil migration and contrast an ad-hoc approach to cell tracking based on interpolation with the model fitting approach we propose in this talk.