Abstract: The Oblomkov–Rasmussen–Shende conjecture relates the Hilbert schemes of a plane curve singularity to a skein-theoretic invariant of its link, called its triply-graded HOMFLYPT homology. Oscar Kivinen and I discovered a more tractable analogue, in which the Hilbert schemes are replaced by other punctual Quot schemes. This led us to conjecture a mysterious motivic substitution relating the two families of varieties. I will survey this research program, as well as current work-in-progress where I expect to prove the full Quot conjecture for all sufficiently generic singularities. If time permits, I will explain what this program tells us about affine Springer fibers and other moduli spaces in representation theory.
Abstract: This talk addresses continuous-time reinforcement learning (RL) in settings where the system dynamics are governed by a stochastic differential equation but remains unknown, with only discrete-time observations available. Existing approaches face fundamental limitations: model-based PDE methods suffer from non-identifiability, while model-free methods based on the classical reinforcement learning framework suffer from large discretization errors by ignoring the continuous-time structure. We introduce Optimal-PhiBE, an equation that integrates discrete-time information into a continuous-time PDE, combining the strengths of both RL and PDE formulations. In linear-quadratic control, Optimal-PhiBE can even achieve accurate continuous-time optimal policy with only discrete-time information. We further develop model-free algorithms for solving Optimal-PhiBE, requiring only minimal modifications to standard RL methods, and we establish convergence guarantees under model misspecification. Unlike classical RL analyses, whose errors typically blow up as the sampling interval shrinks, the approximation error of PhiBE remains stable and independent of discretization by exploiting the smoothness intrinsic to continuous-time dynamics.
Abstract: Machine learning (ML) has recently significantly advanced many domains like natural language processing, computer vision, and speech processing. Since the arrival of ChatGPT in late 2022, generative models have accelerated this rate of progress. ML tools now affect all aspects of the scientific computing endeavor, and themselves can be helped with scientific computing. I will first briefly describe various related themes of research currently underway in my group at UMD: Differentiable Modeling, Implicit Representations, the Attention mechanism in Transformer architectures, and Large Audio Language Models (a recent talk on LALMs). This talk will focus on the use of Differentiable Models and Implicit Neural Representations in scientific modeling. Scientists previously developed forward numerical models encoding domain knowledge. Making these models differentiable allows for this knowledge to be incorporated in deep learning architectures, and allows achieving more efficient computational pipelines for tasks like parameter optimization, inverse problems, and explainable models in data-sparse domains. I will talk about our recent work in a differentiable model for human hearing (with Leslie Famularo & Nishit Anand), room acoustics (Bowen Zhi & Armin Gerami), signal processing for spatial audio (Armin Gerami), computer graphics via Gaussian Splatting and via a regularized SDF formulation (Meenakshi Krishnan), and inverse problems in mathematical physics (Meenakshi Krishnan and Pranav Pulijala).
Abstract: Host genetic structure can significantly alter disease transmission dynamics and long-term disease outcomes. Past work by Beck, Keener, Hoppensteadt, Feng, and others has shown that when pathogen transmission interacts with evolving host traits—such as susceptibility, recovery, or disease-induced mortality—the resulting coupled system can exhibit novel dynamics. These models demonstrated that genetic composition within a host population can shift during an epidemic, and conversely, infection pressures can reshuffle genetic frequencies, producing true feedback between genes and epidemics.
In this talk, I will discuss a specific example of this phenomenon, focusing on the interaction between Plasmodium vivax and the Duffy antigen, a host genetic trait that confers partial protection against infection.
Abstract: Given a compact Kähler manifold, to better understand Mabuchi's K energy we introduce a family of K^beta energies, whose favorable properties are similar to those of the Ding energy from the Fano case. The construction uses Berman's transcendental quantization, and we show that the slope of the K^beta energies along test configurations can be computed using intersection theory. With these ingredients in place we provide a uniform Yau-Tian-Donaldson correspondence that characterizes the existence of a unique constant scalar curvature Kähler metric using test configurations. Combining our techniques with the non-Archimedean approach to K-stability pioneered by Boucksom-Jonsson, we show that the properness of the classical energy can be tested by checking its slope along a distinguished subclass of Li-type models, called log discrepancy models, thus yielding another G-uniform Yau--Tian--Donaldson correspondence. (Joint with Kewei Zhang)
Abstract: Reading a theorem of Puritz from the early 70s through our modern model-theoretic glasses we see a characterisation of tensor products of types in the full theory of the naturals. This (re-)reading was the starting point of joint work (still in progress) with R. Mennuni, in which we extract several candidate definitions generalising the proof of Puritz’s theorem. In the talk I’ll discuss some of these definitions and various contexts in which each of them holds/does not hold.
Abstract: Quantum computing algorithms are naturally suited to simulating quantum dynamics. Many classical dynamical models, obey different physical laws and therefore take very different forms. In this talk I will present a general mapping from linear differential equations d/dt u = L(t) u into the time dependent Schrödinger equation d/dt \psi = -i H(t) \psi, enabling off-the-shelf quantum simulation algorithms to apply. The key idea is a unitary lift: embedding the problem into a slightly larger space where the evolution is unitary, then recover the original solution via a fixed linear functional. The framework has three moving parts—encode, evolve, evaluate—and it is exact whenever simple moment-matching conditions are satisfied. The second half of my talk will present stochastic and nonlinear extensions of this framework.
Abstract: Harmonic maps from a surface to a target manifold are nonlinear analogue of harmonic functions. They form a fundamental class of objects in differential geometry, but most of the time, they are very hard to describe explicitly. In recent years, people have started to study their shape under "typical", "large" or "random" constraints. In this talk, I will give a biased survey of the developments in this field, which connect geometric analysis to dynamical systems and random matrix theory.
Abstract: We'll discuss the paper "Localization for the norm-square of the moment map and the two-dimensional Yang-Mills integral " by C. Woodward in J. Symplectic Geometry, 2005.
Abstract: The stabilized automorphism group of a dynamical system (X,T) is the group of all self-homeomorphisms of X that commute with some power of T. While this is an algebraic object, we show that it captures rich dynamical information. We begin by characterizing the stabilized automorphism groups of odometers and Toeplitz subshifts, establishing an invariance property in these settings. We then extend our results to a broader class of minimal systems, proving that if two such systems have isomorphic stabilized automorphism groups and each has a non-trivial rational eigenvalue, then they must share the same set of rational eigenvalues. We further identify a class of systems for which the assumption of having a non-trivial rational eigenvalue can be removed. Finally, we generalize a known result for mixing shifts of finite type to include all irreducible shifts of finite type.
Abstract: Topological Data Analysis (TDA) is a relatively new field that utilizes foundational results and ideas from algebraic topology and computational geometry to analyze the geometric properties of various types of datasets. All one needs is a dataset with some notion of distance between points (not necessarily a metric), and TDA offers many tools that reveal robust qualitative information about the dataset's structure by, in a sense, 'blurring' the dataset and observing which topological characteristics emerge. In this presentation, I will introduce TDA and the motivation behind it, TDA tools such as Persistent Homology and the Mapper Algorithm, and some applications of TDA to real-world problems in fields such as neuroscience, basketball, and epidemiology. Feel free to join along (no prior knowledge of algebraic topology required) to explore how you can use TDA to blur your eyes and see a clearer picture!
Abstract: The Statistics Program in the Department of Mathematics and the Brin Mathematics Center at the University of Maryland are hosting the inaugural Maryland Statistical Symposium on December 5-6, 2025. This event aims to bring together researchers from the Mid-Atlantic region for research exchange, collaboration, and community building. The broad theme of this year’s symposium is Embracing Classical and Modern Statistics.