The Function-Rips Multifiltration as an Estimator
Abstract
Say we want to view the function-Rips multifiltration as an estimator. Then, what is the target? And what kind of consistency, bias, or convergence rate, should we expect? In this talk I will present on-going joint work with Ethan André (Ecole Normale Supérieure) that aims at laying the algebro-topological ground to start answering these questions.
When we needed a design for our Oxford Mathematics merchandise, we thought we didn't have a badge or coat of arms. Until we realised we did. At the entrance to our building is the Penrose tiling, our mathematical coat of arms.
So our designers, the excellent Stephane Harrison and his team at William Joseph, came up with the idea of the exploding tiles. They have now become our unofficial logo (if you can have such a thing), on all our merchandise and materials.
Matt Clifford, the Prime Minister's Representative at the AI Safety Summit, joins the OXAI Safety and Governance team for a fireside talk.
28th November, Department of Statistics, 6pm.
There will be a chance to talk to Matt after the event as well, as they'll be following the talk with a pizza social. Book now
15:30
The Allen-Cahn equation with weakly critical initial datum
Abstract
Inspired by questions concerning the evolution of phase fields, we study the Allen-Cahn equation in dimension 2 with white noise initial datum. In a weak coupling regime, where the nonlinearity is damped in relation to the smoothing of the initial condition, we prove Gaussian fluctuations. The effective variance that appears can be described as the solution to an ODE. Our proof builds on a Wild expansion of the solution, which is controlled through precise combinatorial estimates. Joint works with Simon Gabriel, Martin Hairer, Khoa Lê and Nikos Zygouras.
15:30
McKean-Vlasov S(P)Des with additive noise
Abstract
Many systems in the applied sciences are made of a large number of particles. One is often not interested in the detailed behaviour of each particle but rather in the collective behaviour of the group. An established methodology in statistical mechanics and kinetic theory allows one to study the limit as the number of particles in the system N tends to infinity and to obtain a (low dimensional) PDE for the evolution of the density of the particles. The limiting PDE is a non-linear equation, where the non-linearity has a specific structure and is called a McKean-Vlasov nonlinearity. Even if the particles evolve according to a stochastic differential equation, the limiting equation is deterministic, as long as the particles are subject to independent sources of noise. If the particles are subject to the same noise (common noise) then the limit is given by a Stochastic Partial Differential Equation (SPDE). In the latter case the limiting SPDE is substantially the McKean-Vlasov PDE + noise; noise is furthermore multiplicative and has gradient structure. One may then ask the question about whether it is possible to obtain McKean-Vlasov SPDEs with additive noise from particle systems. We will explain how to address this question, by studying limits of weighted particle systems.
This is a joint work with L. Angeli, J. Barre, D. Crisan, M. Kolodziejzik.