2:00 Ziheng Wang, EPSRC CDT in Mathematics of Random Systems Student

Continuous-time stochastic gradient descent for optimizing over the stationary distribution of stochastic differential equations

Abstract: We develop a new continuous-time stochastic gradient descent method for optimizing over the stationary distribution of stochastic differential equation (SDE) models. The algorithm continuously updates the SDE model's parameters using a stochastic estimate for the gradient of the stationary distribution. The gradient estimate satisfies an SDE and is simultaneously updated, asymptotically converging to the direction of steepest descent. We rigorously prove convergence of our online algorithm for dissipative SDE models and present numerical results for other nonlinear examples. The proof requires analysis of the fluctuations of the parameter evolution around the direction of steepest descent. Bounds on the fluctuations are challenging to obtain due to the online nature of the algorithm (e.g., the stationary distribution will continuously change as the parameters change). We prove bounds for the solutions of a new class of Poisson partial differential equations, which are then used to analyze the parameter fluctuations in the algorithm.

2:45 Ian Melbourne,  Professor of Mathematics, University of Warwick

Interpretation of stochastic integrals, and the Levy area

Abstract: An important question in stochastic analysis is the appropriate interpretation of stochastic integrals. The classical Wong-Zakai theorem gives sufficient conditions under which smooth integrals converge to Stratonovich stochastic integrals. The conditions are automatic in one-dimension, but in higher dimensions it is necessary to take account of corrections stemming from the Levy area. The first part of the talk covers work with Kelly 2016, where we justified the Levy area correction for large classes of smooth systems, bypassing any stochastic modelling assumptions. The second part of the talk addresses a much less studied question: is the Levy area zero or nonzero for systems of physical interest, eg Hamiltonian time-reversible systems? In recent work with Gottwald, we classify (and clarify) the situations where such structure forces the Levy area to vanish. The conclusion of our work is that typically the Levy area correction is nonzero.

3:45 Break

4:15 Sara Franceschelli, Associate Professor,  École Normale Supérieure de Lyon

When is a model is a good model? Epistemological perspectives on mathematical modelling

When a model is a good model? Must it represent a specific target system? Allow to make predictions? Provide an explanation for observed behaviors?  After a brief survey of general epistemological questions on modelling, I will consider examples of mathematical modelling in physics and biology from the perspective of dynamical systems theory. I will first show that even if it has been little noticed by philosophers, dynamical systems theory itself as a mathematical theory has been a source of questions and criteria in order to assess the goodness of a model (notions of stability, genericity, structural stability). I will then discuss the theoretical fruitfulness of arguments of (in)stability in the mathematical modelling of morphogenesis.

The blow-up B of a scheme X in a closed subscheme Z enjoys the universal property that for any scheme X' over X such that the pullback of Z to X' is an effective Cartier divisor, there is a unique morphism of X' into B over X. It is well-known that the blow-up commutes along flat base change.

In this talk, I will discuss a derived enhancement B' of B, namely the derived blow-up, which enjoys a universal property against all schemes over X, satisfies arbitrary (derived) base-change, and contains B as a closed subscheme. To this end, we will need some elements from derived algebraic geometry, which I will review along the way. This will allow us to construct the derived blow-up as the projective spectrum of the derived Rees algebra, and state its functor of points in terms of virtual Cartier divisors, using Weil restrictions.

This is based on ongoing joint work with Adeel Khan and David Rydh.