Calendar
Posted November 12, 2025
Last modified November 23, 2025
Colloquium Questions or comments?
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 3, 2025
Last modified November 9, 2025
Computational Mathematics Seminar
3:30 pm – 4:30 pm Digital Media Center 1034
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 November 21, 2025
Informal Geometry and Topology Seminar Questions or comments?
1:30 pm Lockett Hall 233
Krishnendu Kar, Louisiana State University
Khovanov Homology
Wrapping up our discussion on Khovanov Homology from this semester.
Posted November 12, 2025
Mathematical Physics and Representation Theory Seminar
1:30 pm – 2:20 pm Lockett 233
Iain Moffatt, Royal Holloway, University of London
Hypermap minors
As mathematicians we conventionally model networks as graphs. In a graph, each edge has exactly two ends, each lying on a vertex. Hypergraphs generalise graphs by allowing an edge to have any number of ends. As the edges of a hypergraph can connect any number of vertices, not just two, they offer a way to model higher-order interactions in networks. Graphs often arise in applications with the additional structure of an embedding in a surface. This is also happens for hypergraphs: a hypermap is a hypergraph embedded in a closed surface. This talk is about hypermaps. I'll begin by reviewing the basics of hypermaps, including various ways to describe them. I'll go on to present a theory of hypermap minors based upon a smoothing operation in cubic graphs. I'll discuss various aspect of this theory such as commutativity, duality and Tutte's triality, polynomials, and relations with Farr's theory of alternating dimaps. This is joint work with Jo Ellis-Monaghan and Steven D. Noble.
Posted November 13, 2025
Last modified November 16, 2025
Colloquium Questions or comments?
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.