| Abstract |
Over the past two decades, gradient flows in the optimal transport framework have found wide-ranging applications in PDEs, probability, and machine learning. However, the long-time behavior of Wasserstein gradient flows for genuinely nonconvex energies remains much less understood.
In geometric analysis, Łojasiewicz-type inequalities provide a powerful framework for studying the asymptotic behavior of gradient flows near critical points. This theory, originating from Łojasiewicz’s work in the 1960s and Simon’s infinite-dimensional extension in 1983, also encompasses questions such as Thom’s gradient conjecture on the asymptotic direction of analytic gradient trajectories.
In this talk, I will discuss a Łojasiewicz theory for nonconvex Wasserstein gradient flows. As applications, we study the long-time behavior of the Keller–Segel equation for chemotactic aggregation and simplified mean-field models of self-attention dynamics motivated by in-context learning. This is joint work in progress with Beomjun Choi (KAIST), Pei-Ken Hung (University of Illinois Urbana-Champaign), and Geuntaek Seo (POSTECH). |
