Georgia Institute of Technology College of Sciences

We introduce in-context operator learning on probability measure spaces for optimal transport (OT). The goal is to learn a single solution operator that maps a pair of distributions to the OT map, using only few-shot samples from each distribution as a prompt and without gradient updates at inference. We parameterize the solution operator and develop scaling-law theory in two regimes. In the nonparametric setting, when tasks concentrate on a low-intrinsic-dimension manifold of source– target pairs, we establish generalization bounds that quantify how in-context accuracy scales with prompt size, intrinsic task dimension, and model capacity. In the parametric setting (e.g., Gaussian families), we give an explicit architecture that recovers the exact OT map in context and provide finite-sample excess-risk bounds. Our numerical experiments on synthetic transports and generative modeling benchmarks validate the framework.

This minicourse provides a friendly, step-by-step introduction to the Kontsevich integral. We begin by demystifying the formula and its construction, showing how it serves as a far-reaching generalization of the classical Gauss linking integral. To establish the invariance of the Kontsevich integral, we explore the holonomy of the Knizhnik–Zamolodchikov (KZ) connection on configuration spaces, utilizing the framework of Chen’s iterated integrals. We will then discuss the universality of the Kontsevich integral for both finite-type (Vassiliev) and quantum invariants, culminating in a concrete combinatorial formula expressed through Drinfeld’s associators. Time permitting, we will conclude by constructing the LMO invariant, demonstrating how it functions as a 3-manifold analog of the Kontsevich integral.

We study robust D-optimal experiment design in generalized linear models, choosing design weights to maximize the worst-case determinant of the Fisher information matrix over a convex parameter uncertainty set. This can be thought of as the natural generalization of D-optimal design in linear regression, and the key challenge is that the information matrix depends on the parameter, so the resulting minimax problem is generally not convex-concave. We show that the desired convexity-concavity, in fact, reduces to a scalar curvature condition on the log-partition function of the exponential family, namely its second derivative h must satisfy the inequality h''h ≥ q(h')² for some q > 1. This insight is connected to the notion of Volumetric Barrier (VB) convexity for self-concordant functions, a result first introduced by Tseng et al (2025) in the context of online quantum state estimation. With self-concordant barriers on the design weights simplex and the parameter uncertainty set, the regularized saddle objective becomes a self-concordant convex-concave (SCCC) function, enabling efficient minimax interior-point methods developed by Nemirovski. We also consider the generalization of the framework, where convexity in the model parameter breaks, but the q-inequality holds up to a deficit proportional to h; in this case, our methods are just as applicable. It turns out that this class includes all canonical GLMs, and we identify logistic regression as the hardest model in the class.

This joint work with Dmitrii Ostrovskii.

We completely characterize the range of $L^p$-boundedness of certain multilinear Radon-like transforms involving vertical projections in the Heisenberg group. This result is now available on arXiv:2603.17147.