Calendar

Posted November 5, 2025
Last modified November 17, 2025

Geometry and Topology Seminar Seminar website

Advika Rajapakse, UCLA
Four Steenrod squares on Khovanov homology

The odd Khovanov spectrum is a space-level link invariant that, after taking (reduced) cohomology, recovers odd Khovanov cohomology. We use the second Steenrod square to disprove several conjectures regarding the odd Khovanov spectrum. We furthermore prove that there exists more than one even Khovanov spectrum, answering another question from Sarkar-Scaduto-Stoffregen.

Posted August 27, 2025
Last modified November 17, 2025

Informal Geometry and Topology Seminar Questions or comments?

Nilangshu Bhattacharyya, Louisiana State University
Khovanov Homology

Continuing our discussion of Khovanov Homology following Melissa Zhang's notes.

Posted November 14, 2025
Last modified November 16, 2025

Colloquium Questions or comments?

Aaron Calderon, University of Chicago
Pants decompositions and dynamics on moduli spaces

Every closed hyperbolic surface X (or Riemann surface or smooth algebraic curve over C) can be described by gluing together pairs of pants (three-holed spheres). Each X can be glued out of pants in many different ways, and Mirzakhani showed that the count of these decompositions is closely related to a certain Hamiltonian flow on the moduli space of hyperbolic surfaces. In the field of Teichmüller dynamics, counting problems on flat surfaces can be related to a different dynamical system on a different moduli space, which, by work of Eskin--Mirzakhani--Mohammadi and Filip, is in turn controlled by special algebraic subvarieties. In this talk, I will survey some of these results and describe a bridge between the two worlds that can be used to transfer theorems between flat and hyperbolic geometry.

Posted November 12, 2025
Last modified November 16, 2025

Colloquium Questions or comments?

Benjamin Zhang, University of North Carolina at Chapel Hill
A mean-field games laboratory for generative artificial intelligence: from foundations to applications in scientific computing

We demonstrate the versatility of mean-field games (MFGs) as a mathematical framework for explaining, enhancing, and designing generative models. We establish connections between MFGs and major classes of flow- and diffusion-based generative models by deriving continuous-time normalizing flows and score-based models through different choices of particle dynamics and cost functions. We study the mathematical structure and properties of each generative model by examining their associated MFG optimality conditions, which consist of coupled forward-backward nonlinear partial differential equations (PDEs). We present this framework as an MFG laboratory, a platform for experimentation, invention, and analysis of generative models. Through this laboratory, we show how MFG structure informs new normalizing flows that robustly learn data distributions supported on low-dimensional manifolds. In particular, we show that Wasserstein proximal regularizations inform the well-posedness and robustness of generative flows for singular measures, enabling stable training with less data and without specialized architectures. We then apply these principled generative models to operator learning, where the goal is to learn solution operators of differential equations. We present a probabilistic framework that reveals certain classes of operator learning approaches, such as in-context operator networks (ICON), as implicitly performing Bayesian inference. ICON computes the mean of the posterior predictive distribution of solution operators conditioned on example condition-solution pairs. By extending ICON to a generative setting, we enable sampling from the posterior predictive distribution. This provides principled uncertainty quantification for predicted solutions, demonstrating how mathematical foundations translate to trustworthy applications in scientific computing.

Posted November 19, 2025

Probability Seminar Questions or comments?

Zhifei Yan, Institute of Basic Science, Korea
Ramsey properties for tilings in random graphs

An $H$-tiling is a collection of vertex-disjoint copies of $H$. In 1975, Burr, Erd\H{o}s and Spencer proved that in every $2$-edge-coloured complete graph $K_n$, the largest monochromatic $H$-tiling has \[\frac{n}{2v(H)-\alpha(H)} - O(1)\] copies of $H$, where $\alpha(H)$ is the independence number of $H$. In this talk, we extend the result of Burr, Erd\H{o}s and Spencer to the random graph $G(n,p)$. We show that for any graph $H$ without isolated vertices, if $p \ge Cn^{-1/m_2(H)}$, then with high probability, every $2$-colouring of $E(G(n,p))$ contains a monochromatic $H$-tiling of size \[\frac{n}{2v(H)-\alpha(H)} - \varepsilon n.\] This also generalizes a classical result of R\"{o}dl and Ruci\'{n}ski [J. Amer. Math. Soc., 1995]: at the threshold where a single monochromatic copy of $H$ is guaranteed, we actually obtain an asymptotically optimal monochromatic $H$-tiling, missing up to an $o(n)$ error term.

Posted July 13, 2025
Last modified November 4, 2025

Control and Optimization Seminar Questions or comments?

Dimitra Panagou, University of Michigan AFOSR YIP, NASA Early Career Faculty, and NSF CAREER Awardee
Safety-Critical Control via Control Barrier Functions: Theory and Applications

This seminar will focus on control barrier functions, as a tool for encoding and enforcing safety specifications, as well as their recent extensions (e.g., robust, adaptive, and predictive) to handle additive perturbations, parametric uncertainty and dynamic environments, with applications to (multi)-robot/vehicle motion planning and coordination. Time permitting, we will also cover how time constraints can be encoded as fixed-time control Lyapunov functions, and the trade-offs between safety and timed convergence.

Posted November 18, 2025

Combinatorics Seminar Questions or comments?

Jake Weber, Louisiana State University
Characterizations of Graph Classes between Claw-free Graphs and Line Graphs of Multigraphs

A line graph $L(G)$ of $G = (V, E)$ is the graph with vertex set E in which $x, y \in E$ are adjacent as vertices if and only if they are adjacent as edges in G. In 1970, Beineke (and Robertson independently) discovered a forbidden induced subgraph characterization for the class of line graphs of simple graphs. Bermond and Meyer in 1973 generalized this characterization to the class of line graphs of multigraphs, denoted $\mathcal{L}$. One such obstruction of these classes is $K_{1,3},$ the claw. In 2008, Chudnovsky and Seymour fully characterized the set of claw-free graphs. In this talk, we present constructive characterizations of classes between $\mathcal{L}$ and claw-free graphs. These constructions aim to provide an alternative approach, with fewer graph classes and operations, to that of Chudnovsky and Seymour. This talk is based on joint work with Guoli Ding.

Posted November 12, 2025
Last modified November 16, 2025

Colloquium Questions or comments?

Colleen Robichaux, University of California, Los Angeles
Deciding Schubert positivity

We survey the study of structure constants in Schubert calculus and its connection to combinatorics and computational complexity.

Posted November 12, 2025
Last modified November 16, 2025

Colloquium Questions or comments?

Keegan Kirk, George Mason University
Nonsmooth Variational Problems, Optimal Insulation, and Digital Twins

How should a fixed amount of insulating material be placed on a heat-conducting body to maximize thermal performance? A thin-shell model of the insulating layer yields, through rigorous asymptotic analysis, a convex but nonsmooth, nonlocal variational problem. To handle the resulting nonsmooth terms, we develop an equivalent Fenchel-dual formulation together with a semi-smooth Newton method built on the discrete duality inherited by Raviart–Thomas and Crouzeix–Raviart elements. We establish a priori and a posteriori error estimates and validate the theory through numerical experiments, including optimal home insulation and spacecraft heat shielding. Beyond its intrinsic mathematical interest, this problem serves as a building block for digital twins, virtual replicas of physical systems that incorporate sensor data and quantify uncertainty to inform decisions about their physical counterparts. One concrete example arises in the refurbishment of a spacecraft’s heat shield after atmospheric re-entry, where available data can be used to infer how much insulation remains on the surface. The model could then optimize where and how much new material to add, under uncertainty about the residual thickness and anticipated thermal loads. The outcome is a high-dimensional, nonsmooth variational problem representative of the optimal control tasks encountered in digital twin settings. The efficient numerical solution of these high-dimensional optimal control problems remains a formidable challenge for the widespread deployment of digital twins. We therefore highlight two complementary research directions aimed at reducing the computational burden: (i) structure aware preconditioning strategies for nonsmooth optimal control problems, including applications to neural network training, and (ii) adaptive tensor-decomposition techniques that enable efficient approximation of high-dimensional stochastic variational problems.

Posted November 3, 2025
Last modified November 9, 2025

Computational Mathematics Seminar

Monika Pandey, Louisiana State University
Adaptive proximal Barzilai–Borwein method for nonlinear optimization

In this presentation, I will discuss adaptive proximal algorithms that builds on the Barzilai–Borwein (BB) stepsize strategy to accelerate gradient-based methods for solving nonlinear composite optimization. For convex problems, we design adaptive rules that automatically adjust the stepsizes using local curvature information, removing the need for traditional line searches, and enhancing both robustness and computational efficiency. These ideas are further extended to nonconvex problems by developing a new nonmonotone line search strategy that preserves global convergence. I will present theoretical guarantees and numerical experiments showing that the proposed Adaptive Proximal Barzilai–Borwein (AdProxBB) method achieves faster convergence and stronger performance than existing proximal gradient algorithms.

Posted August 27, 2025
Last modified October 27, 2025

Informal Geometry and Topology Seminar Questions or comments?

Huong Vo, Louisiana State University
TBD (Independent Talk)

TBD (Independent Talk)