Calendar

Posted August 1, 2025
Last modified November 3, 2025

Control and Optimization Seminar Questions or comments?

Thinh Doan, University of Texas at Austin AFOSR YIP and NSF CAREER Awardee
Multi-Time-Scale Stochastic Approximation as a Tool for Multi-Agent Learning and Distributed Optimization

Multi-time-scale stochastic approximation (SA) is a powerful generalization of the classic SA method for finding roots (or fixed points) of coupled nonlinear operators. It has attracted considerable attention due to its broad applications in multi-agent learning, control, and optimization. In this framework, multiple iterates are updated simultaneously but with different step sizes, whose ratios loosely define their time-scale separation. Empirical studies and theoretical insights have shown that such heterogeneous step sizes can lead to improved performance compared to single-time-scale (or classical) SA schemes. However, despite these advantages, existing results indicate that multi-time-scale SA typically achieves only a suboptimal convergence rate, slower than the optimal rate attainable by its single-time-scale counterpart. In this talk, I will present our recent work on characterizing the convergence complexity of multi-time-scale SA. We develop a novel variant of this method and establish new finite-sample guarantees that achieves the optimal (O(1/k)) convergence rate. Building upon these results, I will also discuss how these advances enable the design of efficient algorithms for key problems in multi-agent learning and distributed optimization over networks.

Posted November 10, 2025

Probability Seminar Questions or comments?

Yangrui Xiang, LSU
Quantitative Hydrodynamics for a Generalized Contact Model

Abstract: We derive a quantitative version of the hydrodynamic limit for an interacting particle system inspired by integrate-and-fire neuron models. More precisely, we show that the L^2-speed of convergence of the empirical density of states in a generalized contact process defined over a d-dimensional torus of size n is of the optimal order O(n^{d/2}). In addition, we show that the typical fluctuations around the aforementioned hydrodynamic limit are Gaussian, and governed by a inhomogeneous stochastic linear equation. This is a joint work with Julian Amorim, Milton Jara.

Posted October 28, 2025
Last modified November 6, 2025

Student Colloquium

Jonathan Walters, Louisiana Tech University
Control Strategies for Flexible Wing Aircraft

Flexible wing aircraft are inspired by nature and are being studied and developed by many major aerospace companies.  In our work, we model small scale aircraft using partial differential equations and employ linear control strategies to shape the wings to a desired target state.  Our work has previously consisted of studying linear controllers such as LQR and LQG applied to our system and studying different damping mechanisms based on material composition. More recently, we've updated our model to incorporate piezo-ceramic patches as realistic controllers and sensors.  An overview of the project and current progress will be presented.

Posted October 30, 2025

Combinatorics Seminar Questions or comments?

Chris Wells, Auburn University
A discrete view of Gromov's filling area conjecture

In differential geometry, a metric surface $M$ is said to be an isometric filling of a metric curve $C$ if $\partial M=C$ and $d_M(x,y)=d_C(x,y)$ for all $x,y\in C$. Gromov's filling area conjecture from 1983 asserts that among all isometric fillings of the Riemannian circle, the one with the smallest surface area is the hemisphere. Gromov's conjecture has been verified if, say, $M$ is homeomorphic to the disk and in a few other cases, but it still open in general. Admittedly, I'm not a differential geometer in even the even the loosest of senses, so we consider instead a particular discrete version of Gromov's conjecture which is likely fairly natural to anyone who studies graph embeddings on arbitrary suraces. We obtain reasonable asymptotic bounds on this discrete variant by applying standard graph theoretic results, such as Menger's theorem. These bounds can then be translated to the continuous setting to show that any isometric filling of the Riemannian circle of length $2\pi$ has surface-area at least $1.36\pi$ (the hemisphere has area $2\pi$). This appears to be the first quantitative lower-bound on Gromov's conjecture that applies to an arbitrary isometric fillings. (Based on joint work with Joe Briggs)

Posted October 15, 2025
Last modified October 16, 2025

Mathematical Physics and Representation Theory Seminar

Paul Sobaje, Georgia Southern University
A Geometric Model For Steinberg Quotients

Let G be a reductive algebraic group over a field of characteristic p > 0. Over the last decade, the longstanding search for a character formula for simple G-modules has been replaced (subsumed even) by the same problem for characters of tilting G-modules. In recent years I began studying "Steinberg quotients" of certain tilting characters. These are formal characters with good combinatorial properties straightforwardly derived from the representation theory of G. In some ways they are also the best candidates to be described by a characteristic p version of Weyl's famous formula. In joint work with P. Achar, we prove that these formal characters are in fact actual characters of a natural class of objects coming from geometric representation theory.

Posted November 12, 2025
Last modified November 13, 2025

Colloquium Questions or comments?

Quanjun Lang, Duke University
Low-Rank Methods for Multitype Interacting Particle Systems and Quantum Superoperator Learning

We introduce a multi-type interacting particle system on graphs to model heterogeneous agent-based dynamics. Within this framework, we develop algorithms that jointly learn the interaction kernels, the latent type assignments, and the underlying graph structure. The approach has two stages: (i) a low-rank matrix sensing step that recovers a shared interaction embedding, and (ii) a clustering step that identifies the discrete types. Under the assumption of the restricted isometry property (RIP), we obtain theoretical guarantees on sample complexity and convergence for a wide range of model parameters. Building on the same low-rank matrix sensing framework, I will then discuss quantum superoperator learning, encompassing both quantum channels and Lindbladian generators. We propose an efficient randomized measurement design and use accelerated alternating least squares to estimate the low-rank superoperator. The resulting performance guarantees follow from RIP conditions, which are known to hold for Pauli measurement ensembles.

Posted November 13, 2025

Informal Analysis Seminar Questions or comments?

Anan Saha, LSU
Learning of Stochastic Differential Equations with integral-drift

Stochastic differential equations (SDEs) with integral drift arise naturally in multiscale systems and in applications where effective dynamics are obtained by averaging over latent or unobserved processes. In such settings, the drift takes the form b̅(x) = ∫ b(x, y) π(dy), with π an unknown probability measure. Our primary goal is the nonparametric estimation of the averaged drift b̅ directly from observable data on X, thereby bypassing the need to recover the unidentifiable measure π, which is of secondary importance for understanding the dynamics of these types of SDE models. In this paper, we develop a nonparametric Bayesian framework for estimating b̅ based on L´evy process priors, which represent π via random discrete supports and weights. This induces a flexible prior on the drift function while preserving its structural relationship to b(x, y). Posterior inference is carried out using a reversible-jump Hamiltonian Monte Carlo (RJHMC) algorithm, which combines the efficiency of Hamiltonian dynamics with transdimensional moves needed to explore random support sizes. We evaluate the methodology on multiple SDE models, demonstrating accurate drift recovery, consistency with stationary distributions, and robustness under different data-generating mechanisms. The framework provides a principled and computationally feasible approach for estimating averaged dynamics in SDEs with integral drift.

Event contact: Laura Kurtz

Posted November 3, 2025
Last modified November 10, 2025

Computational Mathematics Seminar

Jai Tushar, Louisiana State University
Recent Advances in Polytopal Finite Element Methods with Application to Domain Decomposition Methods

Polytopal finite element methods (FEMs) have gained popularity in recent years due to their ability to relax conformity constraints on meshes. This flexibility makes them well-suited for handling complex geometries, adaptive mesh refinement, and coarsening. The design of efficient, robust, scalable solvers for linear systems arising from these kinds of discretizations is important to make them competitive with traditional methods. Non-overlapping domain decomposition methods (DDMs) provide one such family of pre-conditioners. In this talk we first give a heuristic answer to “Why polytopal FEMs?” We then outline several routes from traditional conforming FEMs to polytopal formulations. Next, we present recent tools—rooted in discrete functional analysis and mimicing the continuous setting—that yield condition-number estimates for non-overlapping DDM pre-conditioners arising from these discretizations. Finally, we show robustness and scalability of our pre-conditioner for up to several hundreds of processors performed using the open-source finite element library Gridap.jl.

Posted November 5, 2025

Student Colloquium

Paul Sobaje, Georgia Southern University
Affine Group Schemes and Frobenius Kernels

We will give an introduction to affine group schemes over a field k from the viewpoint of k-group functors.  These objects generalize the notion of an affine algebraic group over k.  One of the most important examples of affine group schemes that are not algebraic groups come from the Frobenius kernels of algebraic groups in characteristic p > 0.  We will discuss these objects and, time permitting, their representation theory.

Posted August 27, 2025
Last modified October 27, 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 5, 2025

Geometry and Topology Seminar Seminar website

Advika Rajapakse, UCLA
TBA

Posted November 12, 2025
Last modified November 13, 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 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 12, 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.