Abstract: This workshop will be focused on various aspects of the Kardar-Parisi-Zhang (KPZ) universality. The latter refers to common random fluctuations of a broad class of probabilistic models, including random growing interfaces, exclusion processes, directed random polymers, last passage percolations, etc. These models exhibit deep connections between various areas of mathematics, such as random matrix theory, stochastic PDEs, combinatorics, and integrable systems. There has been huge progress on the KPZ universality in recent years, and this workshop will help participants keep track of this rapidly developing area of probability theory.
Abstract: Vaughan Jones introduced an index for inclusions of certain von Neumann algebra in the 1980's and proved that it is surpisingly rigid. This rigidity is due to a rich combinatorial structure that is inherent to the representation theory of a subfactor with finite index. Subfactor representations reveal interesting unitary tensor categories, or quantum symmetries, whose algebras of intertwiners always contain the Temperley-Lieb algebras and, if an intermediate subfactor is present, the Fuss-Catalan algebras of Jones and myself. The case of two intermediate subfactors is much more involved and not much progress had been made since the late 1990's.
I will discuss recent work with Junhwi Lim in which we determine the quantum symmetries of a subfactor when two intermediate subfactors occur, and the four algebras form a cocommuting square. These new symmetries turn out to be related to partition algebras and Bell numbers.
Abstract: We discuss several categories of analytic stacks relative to the category of bornological modules over a Banach ring. When the underlying Banach ring is a nonarchimedean valued field, this category contains derived rigid analytic spaces as a full subcategory. When the underlying field is the complex numbers, it contains the category of derived complex analytic spaces. We then consider localising invariants of rigid categories associated to bornological algebras. The main results in this part include Nisnevich descent for derived analytic spaces and a version of the Grothendieck-Riemann-Roch Theorem for derived dagger analytic spaces over an arbitrary Banach ring.
Abstract: In this talk, we present the diffusion-model-based Ensemble Score Filter (EnSF) for accurate and efficient high-dimensional nonlinear filtering that reduces predictive uncertainty in high-dimensional dynamical systems. Nonlinear filtering, also known as data assimilation, is the process of estimating the evolving state of a dynamical system by optimally combining noisy observations with predictions from a numerical model. Conventional particle filters and ensemble Kalman filters lose accuracy in highly nonlinear, large-scale settings. EnSF overcomes this by representing the filtering density via a score-based diffusion model in a pseudo-temporal domain, storing information in the score function rather than finite Monte Carlo samples. A training-free, mini-batch Monte Carlo estimator directly approximates the score function at any pseudo-spatial–temporal location, avoiding costly neural network training while retaining high accuracy. Numerical results on Lorenz-96 systems with up to one million dimensions show EnSF’s substantial gains over the state-of-the-art Kalman type Filter. We further demonstrate the method for data assimilation in calibrating benchmark SPDE solutions and atmosphere–ocean simulation models.
Abstract: Symplectic tiling billiards is a game played on a tiling of the plane that only relies on a notion of parallelism and lends itself to study under the more general non-Euclidean affine structures of the torus. In this talk we will address the classification of affine structures on the torus and how to use them to play symplectic tiling billiards. Demonstrations of the game will be provided in real time along with arguments to explain observed behavior of orbits. This is joint with with Fabian Lander.
Abstract: Recent advances in machine learning have been successful at forecasting complex systems, such as the weather, with purely data-driven models. Here I will describe our group's research into hybrid modeling, combining machine learning with a physics-based model. Our goal is to use data to improve the model's skill both at short-term forecasting and at long-term "climate" simulation. I will include some results from applying the hybrid approach to model the earth's weather and climate.
Abstract: It is known from general relativity that axisymmetric stationary black holes can be reduced to axisymmetric harmonic maps into the hyperbolic plane H^2, while in the Riemannian setting, 4d Ricci-flat metrics with torus symmetry can also be locally reduced to such harmonic maps satisfying a tameness condition. We study such harmonic maps and application includes a construction of infinitely many new complete, asymptotically flat, Ricci-flat 4-manifolds with arbitrarily large second Betti number b_2. Joint work with Song Sun.
Abstract: Which species, communities, and ecosystems will persist under intensifying global change, and which will unravel? I will discuss work that develops ecological and evolutionary theory, individual-based models, and statistical approaches to understand and predict (1) how eco-evolutionary feedbacks shape species invasions and range shifts, (2) how ecological communities adapt or collapse in the face of change, and (3) how the many dimensions of environmental change reshape ecological populations and communities. Mathematical ecology techniques are vital for anticipating and managing ecological change, now more than ever. I would like to encourage applied mathematicians to reach out to ecologists: there is a lot of opportunity and interesting work to be done in this area!
Abstract: Two 20 minute talks. The first talk is AI in Physics.
Our UMD physics department is currently discussing the impact and use of AI/LLMs in research and coursework. I will reflect on my own uses, discussions with students in my courses this past year, and discussions in the faculty committee devoted to this topic.
Abstract: Join us for an inspiring alumni talk hosted by Women in Math, featuring Katharine Gurski, from Howard University, who will share her unique journey from student to her current professional role. This talk offers a chance to gain insight into career possibilities, hear an authentic perspective on navigating the field, and connect with a broader community of women in mathematics. A short reception will follow the talk, providing an opportunity to continue the conversation and network with fellow attendees. All are welcome to attend!
Abstract: For smooth manifolds, the Gysin map of a closed immersion is defined as the cohomology applied on the Pontryagin–Thom collapse map, which collapses the ambient manifold to the one-point compactification of the tubular neighborhood of the closed submanifold. In this talk, I will present a version of the Pontryagin–Thom collapse map in algebraic geometry, more precisely in 𝐏¹-homotopy theory, using a compactified deformation to the normal cone. This yields the Gysin map for all known or unknown cohomology theories with 𝐏¹-homotopy invariance, including étale, Hodge, crystalline, prismatic, etc. I will also survey applications of my Gysin map to more concrete arithmetic questions, including recent work of Carmeli–Feng on Brauer groups of surfaces over finite fields.
Abstract: In 1971, McMullen conjectured a characterization of the face numbers of convex simplicial polytopes. This conjecture, dubbed the “g-conjecture”, was resolved over the following ten years by work of Stanley and Billera–Lee. The extension of this conjecture to simplicial spheres remained open much longer. We will discuss the ingenious characteristic 2 proof given by Papadakis–Petrotou in 2020 and provide a unifying framework for it in commutative algebra. This is joint work in progress with Adiprasito, Oba, Papadakis, and Petrotou.
Abstract: We study Anosov diffeomorphisms on T^3 with a decomposition E^s+E^c+E^u, where E^c expands uniformly. For such systems, u-Gibbs measures are invariant measures whose conditional measures along W^u leaves are absolutely continuous. Our motivating problem is to try to understand when such measures are SRB (conditional measures along W^cu leaves are absolutely continuous). In this talk, I will survey what is known for this problem in the case that E^s and E^u are not jointly integrable (joint work with S. Alvarez, M. Leguil and B. Santiago); and an ongoing work on the jointly integrable case (with S. Crovisier and the aforementioned collaborators). For the jointly integrable case, the proof relies on constructing a "horocycle flow" and studying its ergodic properties.
Abstract: The relativistic Vlasov-Maxwell-Landau system models the dynamics of a dilute hot plasma whose particles interact through collisions and through their self-consistent electromagnetic field. In this talk, we present global regularity estimates for the density and EM field that hold even in the far-from-equilibrium case. This problem features an interesting tug-of-war between the smoothing properties of Landau collisions and the hyperbolic behavior of the Maxwell system.
Abstract: This talk will explore what it means to be a core diagram as well as some of the combinatorics of colored Young diagrams, among other things.