Abstract: We introduce a mixed characteristic analog of F-pure singularities, which we call perfectoid pure singularities. We will present some basic and expected properties of these singularities including their connections with log canonical singularities. We will then show how to produce examples of these singularities via deformation to positive characteristic and discuss some related open questions. This talk is based on joint work with Bhargav Bhatt, Zsolt Patakfalvi, Karl Schwede, Kevin Tucker, Joe Waldron, and Jakub Witaszek.
Abstract: In this talk, I will discuss data-driven methods for kinetic systems, with an emphasis on multiscale modeling and instability control in plasma dynamics. In the first part of the talk, I will introduce our continuous data assimilation algorithm for hydrodynamic moment recovery. Utilizing a relaxation-based nudging system, the method simultaneously reconstructs the solution state and unknown force terms—encoding higher-order moment effects—from sparsely observed data.In the second part, I will discuss dynamical feedback control for the Vlasov–Poisson system. We develop a data-driven control framework using low-rank neural operators, trained via PDE-constrained optimization. The resulting control laws maintain stable performance over a long time horizon. In addition, I will present a cancellation-based control strategy that admits provable infinite-time stabilization.
Abstract: We will present the enumerative min-max theory, which relates the number of genus g minimal surfaces in 3-manifolds to topological properties of the set of all embedded surfaces of genus ≤g. As a consequence, we can show that in every 3-sphere of positive Ricci curvature, there exist ≥5 minimal tori (resolving a conjecture by B. White (1989) in the Ricci-positive case), ≥4 minimal surfaces of genus 2, and ≥1 minimal surface of genus g for all g. This is based on a joint work with Yangyang Li and Zhihan Wang.
Abstract: As systems undergo bifurcations, transient dynamics can provide deep insights into the clockwork of nature and illuminate novel control pathways. I will show how transients involving saddle-node bifurcations reveal the mechanisms of climate change impacts on giant kelp forests, windows of opportunity in restoring ecosystems, and effective solutions to global warming. I will also highlight my lab's new directions to understand network dynamics and point out open mathematical problems associated with each question.
Abstract: Phytoplankton populations are the cornerstone of marine ecosystems, driving atmospheric oxygen production and climate regulation. However, these populations are susceptible to critical density thresholds, where falling below a specific level, the Allee critical value, can precipitate irreversible ecosystem collapse. Addressing this vulnerability, we extend standard oxygen–phytoplankton interaction models by explicitly integrating the Allee effect into both continuous-time and discrete-time frameworks. Through stability and bifurcation analysis, we identify the Allee equilibrium as a strict separatrix; trajectories initiating below this threshold inevitably crash to extinction, while those above possess the potential for coexistence. Beyond classical asymptotic stability, we employ the Resilience Index to quantify the probability of persistence within the basin of attraction and utilize the First Passage Time function to detect signals of critical slowing down. Our simulations uncover a 'ghost of attractor' phenomenon, revealing that the system may linger dangerously in a depletion zone before collapsing. A pivotal insight from this analysis is that systems can be transiently vulnerable to collapse even if they theoretically remain on a chaotic attractor. To further broaden the ecological scope, we introduce an extended model framework that incorporates light limitation and photoinhibition via a Haldane-type growth function. By modeling light attenuation through self-shading, this extension provides a biologically consistent mechanism for density-dependent growth suppression, establishing a robust foundation for future integration of temperature dependence and global warming effects.
Abstract: Let $X$ be a set and let $\mathcal{C}\subset \mathcal{P}(X)$ be a family of subsets of $X$, viewed as possible “concepts” to be learned from labeled samples (i.e., elements of $X$ labeled by $0$ or $1$, depending on whether they belong to some $c^*\in \mathcal{C}$). A class $\mathcal{C}$ is PAC-learnable if there is an algorithm which, given sufficiently many labeled samples, outputs with high probability a subset of $X$ that has a small error with respect to $c^*$. Recent work by Ghazi et.al and Alon et. al. has shown that $\mathcal{C}$ is privately PAC-learnable if and only if it has a finite Littlestone dimension (or, in model-theoretic terms, is stable). This invites a question whether one can exploit additional structure, beyond stability, to obtain better private learning bounds. Following Artem Chernikov’s suggestion, we explore equational classes. I will briefly review the learning-theoretic set up and will describe a natural private learning algorithm for equational classes. This is joint work with Vince Guingona, Miriam Parnes, and Natalie Piltoyan.
Abstract: We present a variational discretization framework for gradient flows based on the Jordan–Kinderlehrer–Otto (JKO) scheme, combined with high-order finite element methods. This approach yields structure-preserving discretizations that inherit key properties of the continuous problem, including energy dissipation and the preservation of bounds for physically relevant quantities such as densities.Building on the dynamic formulation of optimal transport, we reformulate the JKO scheme as a saddle-point problem involving flux variables and Lagrange multipliers, which is well suited for finite element approximation.In the second part, we extend this framework to dynamic optimal transport and mean field control problems. These problems admit a similar variational structure in space–time, leading to a discrete saddle-point system. The resulting formulations can be solved efficiently using scalable first-order methods, such as the primal–dual hybrid gradient (PDHG) algorithm.
Abstract: Zeta functions encode counting problems in geometry, algebra, and arithmetic, and are distinguished by their rich analytic and algebraic structure. In this talk, we consider both local and global zeta functions. Local zeta functions, attached to a prime p or a prime ideal in a number field, are often rational functions in a suitable variable, while global zeta functions combine information over all primes and typically factor as products of local terms. We will survey several different zeta functions that arise in the literature, introduce the Hasse–Weil zeta function, prove some basic properties, and state the Weil conjectures.
Abstract: This talk discusses transportation cost inequalities (TCIs) for the laws of solutions to singular stochastic partial differential equations. A TCI bounds a (generalized) Wasserstein distance in terms of relative entropy and yields concentration of measure on metric spaces. The main technical difficulty is to control the norm of the BPHZ model constructing the solution in terms of a suitable noise norm via a local Hölder-type estimate. To obtain such an estimate, we incorporate an additional integrability structure into the regularity-structure framework and apply a spectral gap inequality. This talk is based on joint work with Ismaël Bailleul (Université de Bretagne Occidentale) and Masato Hoshino (Institute of Science Tokyo).
Abstract: Let $H$ be a smooth and proper Hamiltonian function on $\mathbb{R}^4$. I will discuss a proof that almost every regular level set of $H$ contains at least two closed orbits of the Hamiltonian flow. The proof uses holomorphic curve techniques from symplectic geometry. The techniques also apply, for example, to the problem of finding closed magnetic geodesics on the two-sphere, which I will discuss if there is time to do so.
Abstract: The dynamics for the interface in many free boundary problems is driven by the normal derivative of a corresponding pressure function, and often the free boundary is the boundary of the positivity set of this unknown pressure, which evolves in time. Two well-known examples of these types of models are called the Hele-Shaw equation and the one-phase Muskat problem. In this talk we will describe the set-up of some of these free boundaries and show how in many situations, their solution becomes equivalent to solving a nonlinear fractional heat equation (in one fewer space variables). These fractional heat equations fall into the general scope of what are called Hamilton-Jacobi-Bellman equations, which have enjoyed extensive study in the past 20 years or so (at least for the fractional setting, and much longer for the first and second order settings). Furthermore, many of the well established properties about existence, uniqueness, and regularity for Hamilton-Jacobi-Bellman equations can then be transferred back to the original free boundary problem. We will discuss various recent results in this direction, including global in time well-posedness of solutions for the one-phase Muskat problem, which is joint work with Son Tu and Olga Turanova.