Georgia Institute of Technology College of Sciences

A Brownian motion tree (BMT) model is a Gaussian model whose associated set of covariance matrices is linearly constrained according to common ancestry in a phylogenetic tree. This talk will discuss the complexity of inferring the maximum likelihood (ML) estimator for a BMT model by computing its ML-degree.  The talk will highlight an explicit formula for the ML-degree of the BMT model on a star tree. We will also show that the ML-degree of a BMT model is independent of the choice of the root. This talk is based on work (doi.org/10.1016/j.jsc.2025.102482) with J. I. Coon, S. Cox, & A. Maraj.

Skein lasagna modules are smooth 4-manifold invariants constructed from functorial link homology theories. These invariants are capable of detecting exotic phenomena in dimension 4. Wall-type stabilization questions ask about the behavior of exotic smooth structures under various topological operations. In studying applications of these invariants to Wall-type stabilization problems, we construct an isomorphism between the skein lasagna invariant of a pair of the form (S^2xD^2, L), where L is a link in the boundary S^1xS^2, and the Rozansky-Willis homology of L in S^1xS^2 up to an extra tensor factor. In this talk, we will describe both invariants, describe their relationship, and discuss relevant properties. We will then briefly sketch how the properties of skein lasagna modules are used to establish functoriality for Rozansky-Willis homology and how to upgrade the theory to a new 4-manifold invariant.

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.

Treewidth is a graph parameter commonly used to quantify how "close" a graph is to a tree. Although it is a cornerstone of structural graph theory and algorithm design, it is nearly useless for algorithmic purposes in many dense graph classes. In this talk, we discuss the tree-independence number, a more versatile graph parameter that replaces the standard width measure with the stability number. We will present recent results aimed at characterizing the graph classes in which this parameter enables sub-exponential time algorithms for problems that are, in general, NP-hard.

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.

TBA

Hambly and Jordan (2004) introduced the series-parallel graph, a random hierarchial lattice that is easy to define: Start with the graph consisting of one edge connecting two terminal nodes.  At each subsequent step of the construction, perform independent coin flips for each edge of the graph, and replace the edge by two edges in series if the coin is heads-up or two edges in parallel if tails.  This results in a sequence of random graphs, which can be interpreted as a resistor network.  Hambly and Jordan showed that the logarithm of the resistance grows linearly if the coins are biased to land more often heads-up.  In this talk, I will discuss what happens in the critical case when fair coins are used.  Starting with a new recursive distributional equation (RDE) proposed by Gurel-Gurevich, I develop a framework for analyzing RDE's based on parabolic PDE theory and use this to characterize the asymptotic behavior of the log. resistance.  In the sub- or supercritical case (where the coins are biased), I discuss a tantalizing connection to the Fisher-KPP equation and front propagation.

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.