Calendar

Informal Geometry and Topology Seminar Questions or comments?

3:30 pm – 4:30 pm Monday, October 27, 2025 Lockett 136

Sanjeet Sahoo, LSU
Introduction to Invariant Measures and Ergodicity for Markov Processes

In this talk, we will introduce the concept of transition probability measures and establish criteria for the existence and uniqueness of invariant measures.

Wednesday, October 29, 2025

Posted August 27, 2025
Last modified October 27, 2025

1:30 pm Lockett Hall 233

Nilangshu Bhattacharyya, Louisiana State University
Khovanov Homology

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

Posted September 1, 2025
Last modified October 29, 2025

Control and Optimization Seminar Questions or comments?

3:30 pm Lockett 233

Chen Zhang, Simons Center for Geometry and Physics
Plane Floer homology and the odd Khovanov homology of 2-knots

In this talk, I will discuss joint work with Spyropoulous and Vidyarthi in which we prove a conjecture of Migdail and Wehrli regarding the maps which odd Khovanov homology associates to knotted spheres. Our main tool is the spectral sequence from reduced OKH to Plane Floer homology.

Friday, October 31, 2025

Posted October 7, 2025
Last modified October 9, 2025

9:30 am – 10:20 am Note: First of 2 Seminars for 10/31. Zoom (click here to join)

Alexandre Mauroy, Université de Namur
Dual Koopman Operator Formulation in Reproducing Kernel Hilbert Spaces for State Estimation

The Koopman operator acts on observable functions defined over the state space of a dynamical system, thereby providing a linear global description of the system dynamics. A pointwise description of the system is recovered through a weak formulation, i.e. via the pointwise evaluation of observables at specific states. In this context, the use of reproducing kernel Hilbert spaces (RKHS) is of interest since the above evaluation can be represented as the duality pairing between the observables and bounded evaluation functionals. This representation emphasizes the relevance of a dual formulation for the Koopman operator, where a dual Koopman system governs the evolution of linear evaluation functionals. In this talk, we will leverage the dual formulation to build a Luenberger observer that estimates the (infinite-dimensional) state of the Koopman dual system, and equivalently the (finite-dimensional) state of the nonlinear dynamics. The method will be complemented with theoretical convergence results that support numerical Koopman operator-based estimation techniques known from the literature. Finally, we will extend the framework to a probabilistic approach by solving the problem of moments in the RKHS setting.

Posted October 8, 2025
Last modified October 28, 2025

Control and Optimization Seminar Questions or comments?

10:30 am – 11:20 am Note: Second of 2 Seminars for 10/31. Zoom (click here to join)

Umesh Vaidya, Clemson University
Koopman Meets Hamilton and Jacobi: Data-Driven Control Beyond Linearity

In this talk, we present recent advances in operator-theoretic methods for controlling nonlinear dynamical systems. We begin by establishing a novel connection between the spectral properties of the Koopman operator and solutions of the Hamilton–Jacobi (HJ) equation. Since the HJ equation lies at the core of optimal control, robust control, dissipativity theory, input–output analysis, and reachability, this connection provides a new pathway for leveraging Koopman spectral representations to address control problems in a data-driven setting. In particular, we show how Koopman coordinates can shift the classical curse of dimensionality associated with solving the HJ equation into a curse of complexity that is more manageable through modern computational tools. In the second part of the talk, we discuss safe control synthesis using the Perron–Frobenius operator. A key contribution is the analytical construction of a navigation density function that enables safe motion planning in both static and dynamic environments. We further present a convex optimization formulation of safety-constrained optimal control in the dual (density) space, allowing safety constraints to be incorporated systematically. Finally, we demonstrate the application of this unified operator-theoretic framework to the control of autonomous ground vehicles operating in off-road environments.

Posted October 27, 2025

2:30 pm – 3:30 pm Lockett 233 or click here to attend on Zoom

Matthew Mizell, LSU
Unavoidable minors of matroids with minimum cocircuit size four

When a graph $G$ is a 2-connected and loopless, the set of edges that meet a fixed vertex of $G$ is a bond of $G$ and a cocircuit of its cycle matroid $M(G)$. Because of this, it is common in matroid theory to take minimum cocircuit size as a matroid analog of minimum vertex degree in a graph. Halin and Jung proved in 1963 that every simple graph with minimum degree at least four has $K_5$ or $K_{2,2,2}$ as a minor. In this talk, we will provide a characterization of matroids with minimum cocircuit size four in terms of their unavoidable minors. This talk is based on joint work with James Oxley.

Posted October 21, 2025

Algebra and Number Theory Seminar Questions or comments?

3:30 pm Lockett 232

Michael Lacey, Georgia Institute of Technology
Prime Wiener Wintner Theorem

The classical Wiener Wintner Theorem has an extension to prime averages. Namely, for all measure preserving system $(X,m,T)$, and bounded function $f$ on $X$, there is a set of full measure $X_f\subset X$ so that for all $x\in X_f$, the averages below $$ \frac 1N \sum_{n=1}^N \phi(n) \Lambda (n) f(T^n x ) $$ converge for all continuous $2\pi$ periodic $\phi $. Above, $\Lambda$ is the von Mangoldt function. The proof uses the structure theory of measure preserving systems, the Prime Ergodic Theorem, and higher order Fourier properties of the Heath-Brown approximate to the von Mangoldt function. Joint work with J. Fordal, A. Fragkos, Ben Krause, Hamed Mousavi, and Yuchen Sun.

Saturday, November 1, 2025

Posted October 22, 2025
Last modified October 23, 2025

Conference

until Sunday, November 2, 2025 Lockett Hall 232

Southern Regional Harmonic Analysis Conference

The Southern Regional Harmonic Analysis Conference will focus on current research in harmonic analysis and its applications, featuring plenary talks by Michael Lacey and Irina Holmes. For more details, please refer to conference webpage: https://www.math.lsu.edu/~ha2025/

Event contact: Rui Han, Gestur Olafsson, Naga Manasa Vempati, Fan Yang

Tuesday, November 4, 2025

Posted October 26, 2025
Last modified November 1, 2025

2:00 pm – 3:00 pm Lockett 233 or click here to attend on Zoom

Che-Wei Hsu, National Taiwan University
Hypergeometric Evaluations of L-values and Harmonic Maass Forms

In their earlier work, Bruinier, Ono, and Rhoades asked for an explicit construction of good harmonic Maass forms associated with CM newforms. Building on Ramanujan's theories of elliptic functions to alternative bases, we express $L$-values of certain weakly holomorphic cusp forms in terms of hypergeometric functions. As an application, we construct harmonic Maass forms with explicitly computable Fourier coefficients that are good for specific Hecke eigenforms including certain CM cusp forms.

In this talk, I will briefly review the basic notions of harmonic Maass forms and then present the ongoing joint work with Jia-Wei Guo, Fang-Ting Tu, and Yifan Yang.

Posted October 29, 2025

Informal Geometry and Topology Seminar Questions or comments?

3:30 pm Lockett 136

Christopher Bunting, LSU
Ergodicity of solutions to the stochastic Navier-Stokes equations

The stochastic Navier-Stokes equations has been extensively studied over the past few decades. In this talk, we consider the 2D stochastic Navier-Stokes equations perturbed by an additive noise. We begin by establishing results regarding solutions and provide essential estimates. Using these results, we prove the existence and uniqueness of invariant measure for the solutions of the equations.

Event contact: Laura Kurtz

Wednesday, November 5, 2025

Posted August 27, 2025
Last modified October 27, 2025

1:30 pm Lockett Hall 233

Evan Short, Louisiana State University
Khovanov Homology

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

Posted August 21, 2025
Last modified October 9, 2025

3:30 pm Lockett 233

Matthew Zaremsky, University at Albany (SUNY)
On the Sigma-invariants of pure symmetric automorphism groups

An automorphism of the free group F_n is "pure symmetric" if it sends each generator to a conjugate of itself. The group of all pure symmetric automorphisms of F_n, sometimes called the "McCool group" of F_n, is an interesting and important group with connections to braid groups, motion planning, and mathematical physics. The "Sigma-invariants" of a group are a family of geometric invariants due to Bieri, Neumann, Strebel, and Renz, which are notoriously difficult to compute in general, but reveal a wealth of information about the group and its fibering properties. In recent joint work with Mikhail Ershov, we compute large parts of the Sigma-invariants of the McCool groups, and in particular prove that they are always either empty or dense in the relevant character sphere. One key tool to highlight is an underutilized criterion due to Meinert, which seems likely to have additional future applications.

Thursday, November 6, 2025

Posted August 19, 2025
Last modified November 2, 2025

Control and Optimization Seminar Questions or comments?

3:30 pm Lockett 232

David Roberts, University of Minnesota, Morris
From fewnomials to hypergeometric motives

Understanding the solutions to a given polynomial equation is a central theme in mathematics. In algebraic geometry, one most commonly is focused on solutions in the complex number field $\mathbb{C}$. In number theory, solutions in finite fields $\mathbb{F}_p$ also play an important role.

In this colloquium, I will discuss the case where the given equation has $d+3$ monomials in $d+1$ variables, this being the first generically-behaving case. I will explain how many standard questions about the solutions to these equations in $\mathbb{C}$ and $\mathbb{F}_p$ are concisely and uniformly answered via the theory of hypergeometric motives.

Friday, November 7, 2025

Posted November 4, 2025

Conference

10:30 am – 5:00 pm Lockett 233 and Zoom

Bayou Arithmetic Research Day (BARD 6)

See the event schedule and speakers here: https://bardsmath.com/bard6-schedule/

Posted July 26, 2025

10:30 am – 11:20 am Zoom (click here to join)

Rami Katz, Università degli Studi di Trento, Italy
Oscillations in Strongly 2-Cooperative Systems and their Applications in Systems Biology

The emergence of sustained oscillations (via convergence to periodic orbits) in high-dimensional nonlinear dynamical systems is a non-trivial question with important applications in control of biological systems, including the design of synthetic bio-molecular oscillators and the understanding of circadian rhythms governing hormone secretion, body temperature and metabolic functions. In systems biology, the mechanism underlying such widespread oscillatory biological motifs is still not fully understood. From a mathematical perspective, the study of sustained oscillations is comprised of two parts: (i) showing that at least one periodic orbit exists and (ii) studying the stability of periodic orbits and/or characterizing the initial conditions which yield solutions that converge to periodic trajectories. In this talk, we focus on a specific class of nonlinear dynamical systems that are strongly 2-cooperative. Using the theory of cones of rank k, the spectral theory of totally positive matrices and Perron-Frobenius theory, we will show that strongly 2-cooperative systems admit an explicit set of initial conditions of positive measure, such that every solution emanating from this set converges to a periodic orbit. We further demonstrate our results using the n-dimensional Goodwin oscillator and a 4-dimensional biological oscillator based on RNA–mediated regulation.

Posted October 28, 2025
Last modified November 3, 2025

LSU AWM Student Chapter LSU AWM Student Chapter Website

12:30 pm – 1:30 pm Keiser Lounge

Galen Dorpalen-Barry, Texas A&M
Discussion Session with Dr. Galen Dorpalen-Barry

This is a special AWM-presented discussion session with Dr. Galen Dorpalen-Barry (Texas AM University). It will occur before her talk in the Combinatorics seminar.

Event contact: jgarc86@lsu.edu

Posted October 30, 2025

Applied Analysis Seminar Questions or comments?

2:30 pm – 3:30 pm Lockett 138 or click here to attend on Zoom

Galen Dorpalen-Barry, Texas A&M
Combinatorics and Topology of Conditional Oriented Matroids

Oriented matroids are combinatorial objects that capture much of the topology of (central) real arrangements. A well-know theorem of Salvetti, for example, describes the homotopy type of the complexitied complement of a real arrangement using only the data of its oriented matroid. A conditional oriented matroid plays the role of an oriented matroid when one has a convex body cut by hyperplanes in a real vector space. These arise, for example, in the study of Coxeter arrangements, convex polytopes, and affine arrangements. In this talk, we will give an overview of what’s known about conditional oriented matroids and share new results about their combinatorics and topology. This is a combination of several joint works with various authors including Nick Proudfoot, Jayden Wang, and Dan Dugger.

Monday, November 10, 2025

Posted August 21, 2025
Last modified October 24, 2025

3:30 pm Lockett Hall 233

Roy Goodman, New Jersey Institute of Technology
Leapfrogging and scattering of point vortices

The interaction among vortices is a key process in fluid motion. The n-vortex problem, which models the movement of a finite number of vortices in a two-dimensional inviscid fluid, has been studied since the late 1800s and remains relevant due to its strong link to quantum fluid dynamics. A foundational document in this area is Walter Gröbli's 1877 doctoral dissertation. We apply modern tools from dynamical systems and Hamiltonian mechanics to several problems arising from this work. First, we study the linear stability and nonlinear dynamics of the so-called leapfrogging orbit of four vortices, utilizing Hamiltonian reductions and a numerical visualization method known as Lagrangian descriptors. Second, we analyze the scattering of vortex dipoles using tools from geometric mechanics. While point vortices are typically modeled as massless particles, the final part of this talk will discuss the impact of endowing each particle with a small mass. Although some of the concepts are technical, the presentation will focus on a series of interesting and informative images and animations.

Event contact: Stephen Shipman

Tuesday, November 11, 2025

Posted November 4, 2025

Actuarial Student Association

3:30 pm – 4:30 pm Lockett 136

Phuc Nguyen, Department of Mathematics, Louisiana State University
Capacities, weighted norm inequalities, and nonlinear partial differential equations

I will present a survey of trace inequalities for fractional integrals, highlighting the role of capacities associated to Sobolev spaces and their connections to nonlinear potential theory and nonlinear partial differential equations

Event contact: Laura Kurtz

Posted November 6, 2025

5:30 pm Kessler Lounge (3rd Floor Lockett Hall)

ASA Meeting

We will be joined by Doug and Kaylee from Southern Farm Bureau Insurance. Pizza will be Served

Wednesday, November 12, 2025

Posted August 27, 2025
Last modified October 27, 2025

Informal Geometry and Topology Seminar Questions or comments?

1:30 pm Lockett Hall 233

Matthew Lemoine, Louisiana State University
Topological Data Analysis of Mortality Patterns During the COVID-19 Pandemic (joint work with Megan Fairchild)

This talk will be a brief overview of Topological Data Analysis and will go into some of the work that Megan and I have done. Topological Data Analysis is a relatively new field of study that uses topological invariants to study the shape of data. We analyze a dataset provided by the Centers for Disease Control and Prevention (CDC) using persistent homology and MAPPER. This dataset tracks mortality week-to-week from January 2020 to September 2023 in the United States during the COVID-19 pandemic. We examine the dataset as a whole and break the United States into geographic regions to analyze the overall shape of the data. Then, to explain this shape, we discuss events around the time of the pandemic and how they contribute to the observed patterns.

Posted September 1, 2025
Last modified November 10, 2025

Control and Optimization Seminar Questions or comments?

3:30 pm Locket 233

Jayden Wang, University of Michigan
From Euclid’s first postulate to Lorentzian polynomials

Imagine a world where our basic intuition about points, lines, and planes no longer applies—a world where three points in a three-dimensional linear space need not lie in any plane. This is the tropical world. I will tell a story about tropical linear spaces, where many familiar incidence properties of classical linear geometry fail in surprising ways. I will also discuss how both the fulfillment and the violation of these properties resonate across other areas of mathematics, including algebraic curves, Lorentzian polynomials, and matroid combinatorics.

Friday, November 14, 2025

Posted August 1, 2025
Last modified November 3, 2025

10:30 am – 11:20 am Zoom (click here to join)

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?

11:00 am – 12:00 pm Lockett 233

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

12:30 pm Lockett 138

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

Mathematical Physics and Representation Theory Seminar

2:30 pm – 3:30 pm Zoom

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)

Monday, November 17, 2025

Posted October 15, 2025
Last modified October 16, 2025

1:30 pm – 2:20 pm Lockett 233

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 16, 2025

Algebra and Number Theory Seminar Questions or comments?

4:00 pm 232 Lockett Hall

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.

Tuesday, November 18, 2025

Posted November 15, 2025

2:00 pm – 3:00 pm Lockett 233 or click here to attend on Zoom

Jiaqi Hou, Louisiana State University
Restriction bounds for Maass forms

I will talk about the analytic problem of bounding Hecke–Maass forms. From the general theory of bounding Laplace eigenfunctions on Riemannian manifolds, one obtains local bounds for many different kinds of norms, and these bounds are believed to be far from optimal if the manifold is negatively curved. I will discuss how Hecke–Maass forms on arithmetic hyperbolic 3-folds behave along totally geodesic surfaces and present an improved L^2 bound by the method of arithmetic amplification.

Posted November 3, 2025
Last modified November 10, 2025

Computational Mathematics Seminar

3:30 pm – 4:30 pm Digital Media Center 1034

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

3:30 pm Lockett 276

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 November 13, 2025

3:30 pm – 4:30 pm Room 136

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

Wednesday, November 19, 2025

Posted November 5, 2025
Last modified November 17, 2025

Informal Geometry and Topology Seminar Questions or comments?

1:30 pm Online

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

3:30 pm Lockett Hall 233

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

3:30 pm 232 Lockett Hall

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.

Thursday, November 20, 2025

Posted November 12, 2025
Last modified November 16, 2025

Probability Seminar Questions or comments?

3:30 pm 232 Lockett Hall

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.

Friday, November 21, 2025

Posted November 19, 2025
Last modified November 20, 2025

9:30 am – 10:30 am Zoom

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?

10:30 am – 11:20 am Zoom (click here to join)

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

2:30 pm – 3:30 pm Zoom

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

3:30 pm 232 Lockett Hall

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.

Monday, November 24, 2025

Posted November 12, 2025
Last modified November 23, 2025