Applied Math Colloquium

Abstract: We consider the problem of learning interaction kernels from data for stochastic interacting particle systems and their mean-field limit, described by the McKean–Vlasov PDE. We study two measurement models: observation of a single particle trajectory, and access to discrete noisy space-time measurements of the solution to the mean-field PDE. We treat both the parametric and fully nonparametric settings. We construct estimators based on the spectral analysis of the linearized McKean–Vlasov operator and present a fully Bayesian nonparametric analysis of the problem. Finally, we consider a Bayesian inverse problem for a coupled drift-diffusion–Poisson system, and show how both MCMC methods and the Ensemble Kalman filter can be used to infer the permittivity from noisy measurements of the Fokker–Planck equation.

Abstract: Shifting is a powerful technique in extremal combinatorics initiated by Erdős–Ko–Rado and further developed by Kleitman, Frankl, Füredi, and others. Algebraic shifting is a related algebraic method that I introduced in the early 1980s. It links to a wide array of topics, including convex polytopes, homological properties of simplicial complexes, high-dimensional trees, matroids, graph rigidity, bootstrap percolation, and matrix completion. My first Applied Mathematics Colloquium lecture at MIT in 1983–1984 was on algebraic shifting—a natural meeting point between algebraic and extremal combinatorics (or between Richard Stanley and Daniel Kleitman, if you wish). In this talk, I will discuss several results and open problems surrounding algebraic shifting, highlighting connections to Turán’s (4,3) problem and the Sunflower Conjecture—the focus of the Polymath10 project in 2015–2016. The presentation will be self-contained and accessible.