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Nov 10

Shanyin Tong, University of Pennsylvania

A policy iteration framework for inverse mean-field game

Abstract. Mean-field games (MFGs) model non-cooperative games among large populations of agents and are widely applied in areas such as traffic flow, finance, and epidemic control. Inverse MFGs address the challenge of inferring environmental factors and interaction mechanisms from observed agent behavior. The coupled forward-backward structure of MFG equations makes solving these problems difficult and adds even greater complexity to their inverse problems. In this talk, I will discuss a policy iteration framework for solving inverse MFGs. This approach simplifies the problem by decoupling it into linear PDEs and linear inverse problems, leading to significant computational savings. The framework is flexible, accommodating numerical and machine learning methods for complex geometries and high-dimensional problems. I will also discuss how to combine this framework with operator learning to infer interaction terms. The talk will include convergence results for the policy iteration method, along with numerical examples demonstrating its accuracy and efficiency.

Nov 03

Efe Onaran, University of Pennsylvania

Abstract.
I will introduce a dynamic model of finite, contractible unions of unit cubes without pinches or other local singularities. Its moves add or remove one cube while preserving these constraints, and their permissibility can be checked locally. The dynamics thus resemble Eden growth with an added topological constraint. In dimension two, the admissible states are exactly the finite unions of lattice squares homeomorphic to a disk. We prove a stronger version of the planar case of a question of Pach: every state can be reduced to any prescribed square using only permissible removals, establishing irreducibility of the dynamics. Writing β for the ratio of addition to removal rates and κ for the exponential growth factor of self-avoiding polygons, we prove ergodicity for β < 1/κ, with stationary probability proportional to βⁿ for a state of n squares. We also study excursions from the single-square state that reach size T. After rescaling time, these event times converge in distribution to a homogeneous Poisson point process as T grows. I will also discuss potential applications to Steiner tree sampling and motion planning with rapidly exploring random trees. I will conclude with an extension to distorted disks, lattice models of immersed disks whose boundaries may self-overlap. They provide discrete random-metric-space models with disk topology motivated by Jackiw-Teitelboim gravity. I am developing local-move Markov chains on these objects and studying removability, irreducibility, and ergodic regimes. This is joint work with Yuliy Baryshnikov.

Oct 13

Christian Kuehn, Technical University of Munich

Abstract.
Network dynamics underlie many important phenomena ranging from epidemic spreading and social interactions to neuroscience and engineered systems. Classical mathematical approaches are often tailored to fixed pairwise network structures and become insufficient when networks adapt over time or exhibit higher-order interactions involving multiple nodes simultaneously. This talk presents recent advances in the mathematical analysis of adaptive and polyadic network dynamics, focusing on the development of new theoretical tools that extend beyond standard graph-based frameworks. We discuss adaptive epidemic models, where network topology co-evolves with node states, and highlight results on critical transitions, early-warning indicators, and the role of network measures in predicting qualitative dynamical changes. Furthermore, we introduce models of adaptive social and synaptic plasticity networks that exhibit self-organization near criticality. Finally, we discuss higher-order and polyadic interactions, motivating continuum and mean-field approaches capable of treating temporal, adaptive, and hypergraph-based systems within a unified mathematical framework. These developments provide new perspectives for understanding complex dynamical processes on modern networked systems.

Sept 29

Matthieu Cadiot, École Polytechnique Paris

Overhanging waves in the water wave equations with constant vorticity

Abstract. In this talk, we present a computer-assisted methodology for proving the existence of periodic traveling gravity waves at the free surface of water, in a flow of constant vorticity over a flat bed. Using conformal mappings, the free-boundary problem is reformulated as a quasilinear pseudo-differential equation for a periodic function of a single variable. We then expand the solution in a Fourier series, reducing the problem to an infinite-dimensional system of equations for the Fourier coefficients.

To establish existence, we employ a Newton–Kantorovich type argument, proving the existence of a true solution in a neighborhood of a numerically computed approximation. The verification of the hypotheses of this fixed-point argument relies on a careful combination of analytical estimates and rigorous numerical computations. This approach enables us to rigorously prove the existence of an overhanging wave, that is, wave profiles that are no longer graphs of functions. Moreover, it allows for the construction of branches of solutions parameterized by the relative mass flux, bifurcating from the trivial flat solution. In particular, we rigorously construct a branch that originates at the flat solution and extends to a wave with a self-intersecting profile, passing through overhanging configurations along the way.

Joint work with Susanna Haziot (Princeton).