Georgia Institute of Technology College of Sciences

How close is a random matrix to being singular? This question measures the stability of random linear systems. Classical models treat every direction equally, but in many settings different directions have different scales or variances. We study what happens when a random matrix is deformed by a fixed invertible matrix. Even a diagonal deformation raises a basic question: is stability controlled by the weakest direction, or by the collective geometry of all directions? I will explain why standard estimates miss this geometry and how random subspaces lead to sharper bounds.

The abelian Yang-Mills-Higgs model on (1 + 2)-dimensional Minkowski space is a classical relativistic field theory, where a scalar complex field is coupled to an electromagnetic field. It admits topological soliton solutions called vortices. We present a proof of the asymptotic stability of the degree-one vortex under equivariant perturbations in the self-dual case.

Large language models exhibit a remarkable ability to learn and adapt from context without updating their parameters. In this talk, I will present our recent work on both understanding how foundation models extract task information from context and designing contexts that enable continual improvement during inference. I will first discuss a geometric analysis on in-context learning, providing new insights into how task representations emerge and evolve across layers. I will then discuss how these insights motivate a broader paradigm of inference-time learning, in which context is actively constructed rather than passively consumed. Building on iterative refinement frameworks such as AlphaEvolve, we view context as an evolving memory that stores hypotheses, intermediate solutions, and feedback. Drawing inspiration from optimization and sequential Monte Carlo, we develop principled approaches for designing and updating context over time. Overall, understanding how models read context and how we can systematically write and evolve context may provide a foundation for the next generation of adaptive AI systems.

The multipermutohedral Chow ring was introduced in a series of papers by Clader, Damiolini, Eur, Huang, Li, and Ramadas to study moduli spaces with cyclic symmetry. It generalizes Chow rings of permutohedral varieties and type-B Coxeter arrangements. In this talk, we establish the combinatorial structure of the multipermutohedral Chow ring through an explicit Gröbner basis, yielding a Feichtner-Yuzvinsky-type monomial basis and a formula for the Hilbert series. Using this formula, we refine the palindromicity of the Hilbert series. From a representation-theoretic perspective, we also compute the equivariant Hilbert series under two natural group actions and construct combinatorial maps that establish equivariant unimodality and palindromicity.