CAM Colloquium: Sheehan Olver (Applied Mathematics and Applied Physics, Imperial College, London) - Computing equilibrium distributions of interacting particles

Description

Abstract:
When particles interact, say by attracting or repulsing, they tend to form nice distributions as the number of particles become large. Examples include both physical (electrons in a potential well, space dust) and biological (flocking birds, bacteria). Naïve simulation via differential equations proves insufficient, with computational cost becoming prohibitively expensive in more than one dimensions. Instead, we will introduce techniques based on a measure minimisation reformulation using expansions in weighted orthogonal polynomials to approximate the measures, whereby incorporating the correct singularities of the distributions we can rapidly and accurately compute many such distributions in arbitrary dimensions. This leads to high accuracy confirmation of open conjectures on gap formation (imagine a flock of birds forming a ring, with no density in the middle). These techniques involve understanding the relationship between orthogonal polynomials and singular integral (Hilbert, Riesz, and log kernel) transforms, which have wide reaching consequences. We further explore connections to orthogonal polynomials and random matrix theory, the numerical solution of partial differential equations using boundary integral reformulation, and fractional differential equations.

Bio:
Olver is a Reader in Applied Mathematics and Mathematical Physics at Imperial College, London. He received his Ph.D. from Cambridge in 2008, followed by a Junior Research Fellowship at St. John’s College, Oxford, and a stint at The University of Sydney before moving to Imperial. He works in numerical analysis, with an emphasis on spectral methods using orthogonal polynomials with applications in ordinary and partial differential equations, singular integral equations, and Riemann–Hilbert problems. He was awarded the Adams’ Prize in 2012 for his work on the numerical solution of Riemann–Hilbert problems, and co-authored a SIAM book on the topic. He is an active developer of open-source software, including the ApproxFun.jl package for Julia. His current interests are developing spectral methods for fractional differentiation equations and partial differential equations in multiple dimensions on complicated geometries.