Department Seminars & Colloquia
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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).
We introduce Morse handles on effective oriented smooth four-dimensional orbifolds and explain how invariant splittings determine their indices and ordered handle structures. Even for a fixed isotropy representation and Morse index, different splittings may give nonisomorphic ordered handles. We classify these handles up to orientation-preserving isomorphism by computing normalizer orbits of invariant subspaces for every finite subgroup of SO(4).
We then define surgery on oriented 3-orbifolds by orbifold handle attachment and determine the attaching regions, belt regions, and boundary gluing maps for every ordered handle class of index 1,2, or 3.
These calculations yield an orbifold version of the Lickorish–Wallace theorem. Every closed connected effective oriented smooth 3-orbifold can be transformed into the 3-sphere by a finite sequence of these surgeries. Unlike the manifold case, handles of index 2 alone do not always suffice. We prove that they suffice exactly when there are no tetrahedral, octahedral, or icosahedral vertices. Finally, we determine the minimum total number of handles of indices 1 and 3 from the numbers of these three vertex types.