The seminar meets regularly on Wednesdays at 4pm in Eckhart 202. We also have special seminars during other days. This quarter the seminar will be running remotely via zoom. Access to the zoom links will be provided via the email list. To subscribe or unsubscribe from the email list, you may either go to Camp/PDE email list or contact Cornelia Mihaila.
Many important models in integrable probability (e.g. the KPZ equation, solvable directed polymers, ASEP, stochastic six vertex model) can be embedded into Gibbsian line ensembles. The Gibbs property provides a powerful resampling invariance against Brownian bridges over an arbitrary interval. In this talk, I will explain how to study tightness and path regularity of KPZ line ensemble using this hidden probabilistic structure.
We explore two different problems at the intersection of optimal control and game theory. First we consider models of pedestrian flow, with the goal of guaranteeing that anisotropic interactions in the models are handled in an internally consistent manner. We focus on two classes of models, one based on inter-crowd interactions and another based on intra-crowd interactions. For both cases, we prove sufficient conditions under which the direction of motion is determined uniquely almost everywhere in the domain. For the inter-crowd models, we also prove sufficient conditions under which the Nash Equilibrium is guaranteed to be unique. Next we formulate control-theoretic models of environmental crime in protected areas, such as wildlife poaching or illegal logging. We carefully model the optimal control problem faced by extractors aiming to illegally extract resources from a protected area such as a national park. The extractors are assumed to rationally choose their paths in and out of the protected area in order to maximize their expected payoffs, balancing the resources obtained from extraction with their travel time and risk of detection by authorities.
A basic fact of geometry is that there is no length-preserving map from any neighborhood of a sphere to the plane. But what happens if you force a thin elastic sphere, which likes to deform in an approximately isometric way, to reside nearby a plane? It wrinkles, and the wrinkles form a pattern we'd like to understand. This talk will present a recently derived method based on energy minimization for determining the wrinkle patterns formed when a curved shell is put onto a soft substrate. Perhaps surprisingly, the energetically optimal patterns can in many cases be solved for exactly and by hand. Further investigation of the solutions leads to a beautiful and unexpected connection between the patterns of negatively and positively curved shells. These theoretical results are obtained via a Gamma-convegence argument, and through convex analysis of the limiting problem. The predicted patterns match the results of numerous experimental and numerical tests, done in collaboration with Eleni Katifori (U. Penn) and Joey Paulsen (Syracuse).
In this talk we will present a generalization of the theory of propagation of chaos to backward (weakly) interacting diffusions. The focus will be on cases allowing for explicit convergence rates and concentration inequalities in Wasserstein distance for the empirical measures. As the main application, we derive results on the convergence of large population stochastic differential games to mean field games, both in the Markovian and the non-Markovian cases. The talk is based on joint works with M. Laurière and Dylan Possamaï.