Calendar
Posted November 11, 2024
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3:30 pm Lockett 232
Benjamin Dodson, Johns Hopkins University
Global well-posedness and scattering for the radial, conformal wave equation
In this talk we prove global well-posedness and scattering for the radially symmetric nonlinear wave equation with conformally invariant nonlinearity. We prove this result for sharp initial data.
Posted January 13, 2025
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3:30 pm – 4:30 pm Lockett 232
Trinh Tien Nguyen, University of Wisconsin Madison
Boundary Layers in Fluid Dynamics and Kinetic Theory
Abstract: In this talk, I will discuss recent results on Prandtl boundary layer theory in fluid dynamics. We demonstrate that the Prandtl expansion holds for initial data that is analytic near the boundary under the no-slip boundary condition. I will then present a recent result on the validity of the Prandtl expansion from Boltzmann theory, marking an important step toward justifying other types of approximate solutions (arising from fluid dynamics) as macroscopic limits of the kinetic Boltzmann equations.
Posted January 10, 2025
Last modified January 17, 2025
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3:30 pm – 4:30 pm Lockett 232
Suhan Zhong, Texas A&M University
Polynomial Optimization in Data Science
Abstract: Optimization plays a pivotal role in data science. Recent advances in polynomial optimization have introduced innovative methods to solve many challenging problems in this field. In this talk, I will showcase the application of polynomial optimization through the lens of two-stage stochastic models. Additionally, I will provide a brief overview of the underlying theory and discuss potential future research directions.
Posted January 15, 2025
Last modified January 21, 2025
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3:30 pm – 4:30 pm Lockett 232
Saber Jafarpour, University of Colorado Boulder
Safety and Resilience of Learning-enabled Autonomous Systems: A Monotone Contracting System Perspective.
Abstract: Learning-enabled autonomous systems are increasingly deployed for decision-making in safety-critical environments. Despite their substantial computational advantages, ensuring the safety and reliability of these systems remains a significant challenge due to their high dimensionality and inherent nonlinearity. In this talk, we leverage tools and techniques from control theory to develop theoretical and algorithmic methods for certifying the safety and robustness of learning-enabled autonomous systems. Our approach investigates safety and resilience from a reachability perspective. We employ contraction and monotone systems theories to develop computationally efficient frameworks for approximating reachable sets of autonomous systems. We demonstrate how these frameworks can be applied to verify and train robust standalone neural networks and to provide run-time safety assurance in systems with learning-based controllers.
Posted December 5, 2024
Last modified January 22, 2025
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3:30 pm Lockett 232
Ken Ono, University of Virginia
Partitions detect primes
This talk presents “partition theoretic” analogs of the classical work of Matiyasevich that resolved Hilbert’s Tenth Problem in the negative. The Diophantine equations we consider involve equations of MacMahon’s partition functions and their natural generalizations. Here we explicitly construct infinitely many Diophantine equations in partition functions whose solutions are precisely the prime numbers. To this end, we produce explicit additive bases of all graded weights of quasimodular forms, which is of independent interest with many further applications. This is joint work with Will Craig and Jan-Willem van Ittersum.
Posted January 28, 2025
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3:30 pm – 4:30 pm Lockett 232
Federico Glaudo, Institute for Advanced Study, Princeton
A Journey through PDEs and Geometry
This talk will explore a range of intriguing questions that lie at the crossroads of partial differential equations and geometry. Topics include the stability of near-solutions to PDEs, the isoperimetric inequalities on curved spaces, as well as the random matching problem. The aim is to make the ideas accessible and engaging for a broad mathematical audience.
Posted February 5, 2025
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2:30 pm – 3:30 pm Zoom
Ajay Chandra, Imperial College London
An Invitation to Singular Stochastic Partial Differential Equations
Abstract: In this talk I will start by motivating the fundamental importance of singular stochastic partial differential equations in (i) our understanding of the large-scale behaviour of dynamic random systems and (ii) developing a rigorous approach to quantum field theory. I will describe the key mathematical difficulties these equations pose, and sketch how combining analytic, probabilistic, and algebraic arguments have allowed mathematicians to overcome these difficulties and develop a powerful new PDE theory. I’ll also discuss some more recent developments in this area, namely applications to gauge theory and non-commutative probability theory.
Posted February 19, 2025
Last modified March 10, 2025
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3:30 pm – 4:30 pm Lockett 232
Justin Holmer, Brown University
Dynamics of Solitary Waves
Solitary waves arise as exact coherent structures in a range of nonlinear wave equations, including the nonlinear Schrödinger, Korteweg–de Vries, and Benjamin–Ono equations. These equations have broad applications in areas such as water wave theory, plasma physics, and condensed matter physics. When certain types of perturbations are introduced, the solitary wave retains its overall form while its shape and position adjust to accommodate the new conditions. In this talk, I will present some theoretical results on the modulation of solitary wave profiles under such perturbations, supported by numerical simulations that illustrate and validate these findings.
Posted October 2, 2025
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3:30 pm Lockett 232
Wilhelm Schlag, Yale University
On uniqueness of excited states and related questions
This talk will present the long-standing problem of excited states uniqueness for the nonlinear Schroedinger equation. We will describe the history of the problem, it's relevance to long-term dynamics of nonlinear wave equations, related spectral problems, and progress on the uniqueness question via rigorous numerics. The recent breakthrough by Moxun Tang, who found an analytical proof, will be discussed.
Posted October 6, 2025
Last modified October 13, 2025
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3:30 pm Lockett 232
Paul Kirk, Indiana University
On the SU(2) character variety of a closed oriented genus 2 surface
A celebrated theorem of Narasimhan-Ramanan asserts that the singular variety $X(F_2)=Hom(\pi_1(F_2),SU(2))/Conjugation$ is homeomorphic to $CP^3$. The proof passes through the (mysterious) Narasimhan-Seshadri correspondence. I'll outline an elementary differential topology proof that $X(F_2)$ is a manifold, homeomorphic to CP^3, and discuss how 3-manifolds with genus 2 boundary determine embedded lagrangians in $X(F_2)$. If time permits, I'll end the talk with a discussion of context, particularly with a program known as the Atiyah-Floer conjecture.
Posted October 21, 2025
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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.
Posted August 19, 2025
Last modified November 2, 2025
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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.
Posted November 12, 2025
Last modified November 16, 2025
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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.
Posted November 14, 2025
Last modified November 16, 2025
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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.
Posted November 12, 2025
Last modified November 16, 2025
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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.
Posted November 12, 2025
Last modified November 16, 2025
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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.
Posted November 12, 2025
Last modified November 16, 2025
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4:00 pm 232 Lockett Hall
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 13, 2025
Last modified November 16, 2025
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3:30 pm 232 Lockett Hall
Sky Cao, Massachusetts Institute of Technology
Yang-Mills, probability, and stochastic PDE
Originating in physics, Yang-Mills theory has shaped many areas of modern mathematics. In my talk, I will present Yang-Mills theory in the context of probability, highlighting central questions and recent advances. In particular, I will discuss the role of stochastic partial differential equations (SPDEs) in these developments and survey some of the recent progress in this field.
Posted November 13, 2025
Last modified November 17, 2025
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3:30 pm 232 Lockett Hall
Mengxuan Yang, Princeton University
Flat bands in 2D materials
Magic angles are a hot topic in condensed matter physics: when two sheets of graphene are twisted by these angles, the resulting material is superconducting and the so-called energy bands are flat and topological. In 2011, Bistritzer and MacDonald proposed a model that is experimentally very accurate in predicting magic angles. In this talk, I will introduce some recent mathematical progress on the Bistritzer--MacDonald's model, including the mathematical characterization of magic angles and flat bands, the generic existence of Dirac cones and how topological phase transitions occur at magic angles. I will also discuss some new mathematical discoveries in twisted multilayer graphene.
Posted November 13, 2025
Last modified November 16, 2025
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3:30 pm 232 Lockett Hall
Peter Bradshaw, University of Illinois Urbana-Champaign
To be announced
Posted November 12, 2025
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3:30 pm Lockett 232
Iain Moffatt, Royal Holloway, University of London
Graphs in surfaces, their one-face subgraphs, and the critical group
Critical groups are groups associated with graphs. They are well-established in combinatorics; closely related to the graph Laplacian and arising in several contexts such as chip firing and parking functions. The critical group of a graph is finite and Abelian, and its order is the number of spanning trees in the graph, a fact equivalent to Kirchhoff’s Matrix--Tree Theorem.
What happens if we want to define critical groups for graphs embedded in surfaces, rather than for graphs in the abstract?
In this talk I'll offer an answer to this question. I'll describe an analogue of the critical group for an embedded graph. We'll see how it relates to the classical critical groups, as well as to Chumtov's partial-duals, Bouchet's delta-matroids, and a Matrix--quasi-Tree Theorem of Macris and Pule, and describe how it arises through a chip-firing process on graphs in surfaces.
This is joint work with Criel Merino and Steven D. Noble.
Posted November 13, 2025
Last modified November 16, 2025
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3:30 pm 232 Lockett Hall
Sean Cotner, University of Michigan
Propagating congruences in the local Langlands program
The Langlands program is a vast generalization of quadratic reciprocity, aimed at understanding the algebraic field extensions of the rational or p-adic numbers. In this talk, I will describe a biased and incomplete history of the classical local Langlands program; recent developments in making it categorical, integral, and modular; and joint work-in-progress with Tony Feng concerned with patching together the modular theory to understand the classical theory.