Tue, 13 May 2025
16:00
L6

Random matrix theory and optimal transport

Bence Borda
(University of Sussex)
Abstract

The Wasserstein metric originates in the theory of optimal transport, and among many other applications, it provides a natural way to measure how evenly distributed a finite point set is. We give a survey of classical and more recent results that describe the behaviour of some random point processes in Wasserstein metric, including the eigenvalues of some random matrix models, and explain the connection to the logarithm of the characteristic polynomial of a random unitary matrix. We also discuss a simple random walk model on the unit circle defined in terms of a quadratic irrational number, which turns out to be related to surprisingly deep arithmetic properties of real quadratic fields.

Tue, 11 Feb 2025
16:00

Derivative moments of CUE characteristic polynomials and the Riemann zeta function

Nick Simm
(University of Sussex)
Abstract
I will discuss recent work on the derivative of the characteristic polynomial from the Circular Unitary Ensemble. The main focus is on the calculation of moments with values of the spectral parameter z inside the unit disc. We investigate three asymptotic regimes depending on the distance of z to the unit circle, as the size of the matrices tends to infinity. I will also discuss some corresponding results for the derivative of the Riemann zeta function. This is joint work with Fei Wei (Sussex).



 

Tue, 14 Jun 2022

12:00 - 13:15
Virtual

Quantum hair and black hole information

Xavier Calmet
(University of Sussex)
Abstract

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.

Tue, 09 Jun 2020

15:30 - 16:30

Characteristic polynomials of non-Hermitian matrices, duality, and Painlevé transcendents

Nick Simm
(University of Sussex)
Abstract

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).

Thu, 30 Nov 2017

14:00 - 15:00
L4

Error analysis for a diffuse interface approach to an advection-diffusion equation on a moving surface

Dr Vanessa Styles
(University of Sussex)
Abstract

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.

Thu, 12 Oct 2017

14:00 - 15:00
L4

A robust and efficient adaptive multigrid solver for the optimal control of phase field formulations of geometric evolution laws with applications to cell migration

Professor Anotida Madzvamuse
(University of Sussex)
Abstract

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.

Mon, 23 Jan 2017

16:00 - 17:00
L4

Linearisation of multi-well energies

Mariapia Palombaro
(University of Sussex)
Abstract

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.) 

Thu, 28 May 2015

14:00 - 15:00
L5

Semi-Langrangian Methods for Monge-Ampère Equations

Dr Max Jensen
(University of Sussex)
Abstract

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.

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