ON DEMAND PDE CDT Students & Alumni Reunion Event - DAY TWO
Nikolaos Kolliopoulos, Carnegie Mellon University
Cohort 1 Alumni
Propagation of chaos for maxima of particle systems with mean-field drift interaction.
We present a strong propagation of chaos result for the largest particle of a finite system with correlation due to a mean-field term in the drifts of the Stochastic Differential Equations (SDEs) describing the evolution of the particles. At any fixed time and under the appropriate centering and scalling, it is shown that the value of the largest particle of our system has the same asymptotic behaviour as that of a system of several independent copies of a solution to the corresponding McKean-Vlasov SDE when the sizes of both systems tend to infinity. This allows for the asymptotic distribution of the normalized value of the largest particle to be determined by using results from standard Extreme Value Theory. Our work is motivated by the need to study the top performing assets in a large stochastic portfolio.
Miles Caddick, Highgate School
Cohort 1 Alumni
Teaching after a DPhil
Three weeks after submitting my DPhil thesis I was teaching maths to a classroom of secondary school pupils, not really knowing what I was doing. A year and a half later I was stuck inside my flat teaching maths to secondary school pupils via Zoom, again not really knowing what I was doing. In this talk I will discuss: why and how I got into teaching and my experience of learning on the job; how I adapted my teaching during two periods of lockdown; and how this has informed some of my practice going forward.
Dr Ioannis Papadopoulos, Imperial College London
Cohort 4 Alumni
A sparse spectral element method for a one-dimensional fractional Helmholtz-like problem
Fractional partial differential equations (FPDEs) are used to model nonlocal processes such as power-law absorption in acoustics or Lévy flights (a random walk in which the probability distribution for the step length is heavy tailed). Unlike traditional PDEs, the support of solutions to FPDEs does not necessarily live in the support of the data. Moreover, singularities can occur in solutions even with smooth data and domains. This presents a considerable problem for numerical solvers. We will outline a sparse spectral element method for the one-dimensional fractional Helmholtz equation (which can also include Hilbert transform and derivative terms).
Prof. Grigory Seregin - Mathematical Institute, University of Oxford
Emeritus - Oxford Centre for Nonlinear PDEs
Criticality and related questions in the theory of the Navier-Stokes equations.
In the talk, I am going to discuss some results of my former PDE CDT students: Tobias Barker and Francis Hounkpe.