Georgia Institute of Technology College of Sciences

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.

In 1986, Thurston introduced a norm on the first cohomology of a 3-manifold $M$ and showed that it can be used to study which cohomology classes are induced by a fibration of $M$ over the circle. In 1998, McMullen introduced a norm on first cohomology that depends only on the Alexander polynomial and showed that it provides a lower bound for the Thurston norm. In this talk, we will introduce the Thurston and Alexander norms and explain why there is an inequality relating the two. To do this, we will define the Alexander polynomial in terms of elementary ideals, and we will use this perspective to understand how topological information is encoded in the exponents of the Alexander polynomial.

In this talk I will outline some computations and applications involving Khovanov and Lee skein lasagna modules, including the detection of some exotic pairs of 4-manifolds.  This work is joint with Qiuyu Ren.  If time allows I will also discuss a new version of the lasagna module which should ease the computational complexity for manifolds with 1-handles.  This work is joint with Qiuyu Ren, Ian Sullivan, Paul Wedrich, and Melissa Zhang.