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.
Mean-field game theory borrows ideas from statistical physics to provide a tractable approximation of very large multi-agent systems. Applications are ubiquitous in today's highly interconnected world, from crowd motion to macroeconomics and distributed robotics. Real-world problems often lead to models which are in high dimension or not fully specified, hence a recent surge of interest for the question of computing solutions with mesh-free and model-free methods. In this talk, we will mainly focus on the question of learning in mean-field games, in cooperative or non-cooperative settings.
Since their introduction around 2006, mean field games (MFGs) have been extensively studied and attracted the interest of many researchers from different backgrounds due to both their interesting mathematical properties and their applicability in a diversity of contexts. This talk will focus on some recent MFG models motivated by crowd motion, called minimal-time MFGs, in which agents want to minimize the time required to reach a given target set. In order to model congestion, the maximal speed of an agent is assumed to depend on the distribution of other agents around their position. After briefly presenting MFGs in general and minimal-time MFGs, the talk will review some recent results in two situations, corresponding to first-order MFGs, in which agents follow deterministic trajectories, and second-order MFGs, in which agents' trajectories are perturbed by additive Brownian motions, assumed to be mutually independent. We will present results on the existence of equilibria, their characterization, their asymptotic behavior, and their regularity properties. We will also discuss recent results which suggest how to tackle the case of minimal-time MFGs with state constraints. This talk is based on joint works with Romain Ducasse, Samer Dweik, Saeed Sadeghi Arjmand, and Filippo Santambrogio.
The question of uniqueness of general bounded vorticity solutions to the 2D Euler equations on singular domains is a challenging one because these solutions are typically not close to Lipschitz near boundary singularities. In this talk I will present a general sufficient condition on the geometry of the domain that guarantees global uniqueness for all weak solutions whose vorticities are bounded and initially constant near the boundary. This condition is only slightly more restrictive than exclusion of corners with angles greater than $\pi$ and, in particular, is satisfied by all convex domains. A crucial ingredient of the proof is showing that vorticities of such solutions remain constant near the boundary because fluid particle trajectories starting inside the domain cannot reach it in finite time. The latter also holds for general bounded vorticity solutions on domains satisfying our sufficient condition, and the condition is in fact sharp in this sense as particle trajectories on domains that are arbitrarily close to satisfying it can reach their boundaries in finite time.
Models arising in biology are often written in terms of Ordinary Differential Equations. The celebrated paper of Kermack-McKendrick (1927), founding mathematical epidemiology, showed the necessity to include parameters in order to describe the state of the individuals as time elapsed after infection. During the 70s, many mathematical studies where developed when equations are structured by age, size or a physiological trait. The talk will present some structured equations, show that a universal relative entropy structure is available in the linear case. In the nonlinear cases it might be that periodic solutions occur, which can be interpreted, e.g., as network activity in the neuroscience. When the equations are conservation laws, the Monge-Kantorovich distance, or some variants, also give a general control of solutions.