Georgia Institute of Technology College of Sciences

Rescaling limits were first introduced by Jan Kiwi to study degenerations of rational maps of degree at least two. Building on the work of Luo and Favre–Gong, we explain how rescaling limits can serve as a substitute for a good compactification of $Rat_d$, the moduli space of degree d rational maps. In particular, this framework allows one to promote pointwise results to uniform statements in a systematic way. 

I'll report on an ongoing project, partly joint work with J. Hillman, aimed at finding criteria for the existence of sections on a given Lefschetz fibration over a surface. We will start by presenting a nice algebraic criterion for the existence of sections in a surface bundle and explain what goes wrong if we try to apply it to the more general Lefschetz fibration case. The question of when a nullhomotopic loop in the boundary of a Lefschetz fibration over the disk can be extended to a section over the whole disk is one such subtle issue. Our computations suggest that working with continuous or smooth sections leads to different answers.

It is well-known that a spacetime which expands sufficiently fast can stabilize the fluid for relativistic/Einstein-fluid systems. One may wonder whether the expansion of the fluid, instead of the background spacetime geometry, is also able to achieve a similar stabilizing effect. As an attempt to address this question, we consider the free boundary relativistic Euler equations in Minkowski background M1+3 equipped with a physical vacuum boundary, which models the motion of relativistic gas. For the class of isentropic, barotropic, and polytropic gas, we construct an open class of initial data which launch future-global solutions. Such solutions are spherically symmetric, have small initial density, and expand asymptotically linearly in time. In particular, the asymptotic rate of expansion is allowed to be arbitrarily close to the speed of light. Therefore, our main result is far from a perturbation of existing results concerning the classical Euler counterparts. This is joint work with Marcelo Disconzi and Chenyun Luo.

We introduce proximal optimal transport divergences that provide a unifying variational framework interpolating between classical f-divergences and Wasserstein metrics. From a gradient-flow perspective, these divergences generate stable and robust dynamics in probability space, enabling the learning of distributions with singular structure, including strange attractors, extreme events, and low-dimensional manifolds, with provable guarantees in sample size.

We illustrate how this mathematical structure leads naturally to generative particle flows for reconstructing nonlinear cellular dynamics from snapshot single-cell RNA sequencing data,including real patient datasets, highlighting the role of proximal regularization in stabilizing learning and inference in high dimensions.

Bio: Markos Katsoulakis is a Professor of Applied Mathematics and an Adjunct Professor of Chemical & Biomolecular Engineering at UMass Amherst,  whose research lies at the interface of PDEs, uncertainty quantification, scientific machine learning, and information theory. He serves on the editorial boards of the SIAM/ASA Journal on Uncertainty Quantification, the SIAM Journal on Scientific Computing, and the SIAM Mathematical Modeling and Computation book series. He received his Ph.D. in Applied Mathematics from Brown University and his B.Sc. from the University of Athens. His work has been supported by AFOSR, DARPA, NSF, DOE, and the ERC.