Calendar

Posted November 12, 2025
Last modified November 16, 2025

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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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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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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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Peter Bradshaw, University of Illinois Urbana-Champaign
To be announced

Posted November 12, 2025

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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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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.