Sep 12: Paul Dupuis (Brown University)

Title: Robust control of a risk-sensitive performance measure

Abstract: Using the variational formula that relates relative entropy and exponential integrals, one can formulate an optimization problem whose solution provides controls that are min/max optimal for a family of models and with respect to an ordinary performance measure. The family can be viewed as a perturbation of a nominal or design model, with distances measured by relative entropy. In this talk, we describe a generalization to the situation where the performance measure is a risk-sensitive functional, i.e., a quantity largely determined by tail properties. The generalization is based on variation formula involving Renyi divergence. To illustrate we consider the problem of optimal scheduling for an uncertain queueing system.

Sep 26: Subhabrata Sen (Harvard)

Title: Sampling convergence for random graphs: graphexes and multigraphexes

Abstract: We will look at structural properties of large, sparse random graphs through the lens of sampling convergence (Borgs, Chayes, Cohn and Veitch '17). Sampling convergence generalizes left convergence to sparse graphs, and describes the limit in terms of a graphex. We will introduce this framework and motivate the components of a graphex. Subsequently, we will discuss the graphex limit for several well-known sparse random (multi)graph models.

This is based on joint work with Christian Borgs, Jennifer Chayes, and Souvik Dhara.

Oct 17: Nike Sun (MIT)

Title: Capacity lower bound for the Ising perceptron

Abstract: The perceptron is a toy model of a simple neural network that stores a collection of (randomly) given patterns. Its analysis reduces to a simple problem in high-dimensional geometry, namely, understanding the intersection of the cube (or sphere) with a collection of random half-spaces. Although a heuristic analysis of the model was successfully completed by physicists in the 1980s, it remains open to develop a mathematical understanding. I will summarize what is known, and present some recent progress. This talk is based on joint work with Jian Ding.

Nov 14: Pierre Nyquist (KTH)

Title: Continuum limit of a hard-sphere particle system by large deviations.

Abstract: Many stochastic particle systems have well-defined continuum limits: as the number of particles tends to infinity, the density of particles converges to a deterministic limit that satisfies a partial differential equation. In this talk I will discuss a specific example of this: a system consisting of particles with finite size. In two and three dimensions they are spheres, in one dimension rods. The particles move by Brownian noise and cannot overlap with each other, leading to a strong interaction with neighbouring particles. Previous studies include numerical simulations and formal asymptotic results, along with conjectures on the limit, but no rigorous results.

We will consider the one-dimensional setting and a scaling in which the number of particles tend to infinity while the volume fraction of the rods remain constant. Using large deviations for empirical measures we give a complete picture of the convergence of the particle system and derive the gradient flow structure for the limit evolution. The latter gives clear interpretations for the driving functional and the dissipation metric and how they derive from the underlying particle system.

This is based on joint work with Nir Gavish and Mark Peletier.