Georgia Institute of Technology College of Sciences

Deligne connects the weight-zero compactly supported cohomology of a complex variety to the combinatorics of its compactifications. In this talk, we use this to study the moduli space of n-marked hyperelliptic curves. We use moduli spaces of G-admissible covers and tropical geometry to give a sum-over-graphs formula for its weight-0 compactly supported Euler characteristic, as a virtual representation of S_n. This is joint work with Melody Chan and Siddarth Kannan.

Real Heegaard Floer homology is a new invariant of branched double covers, introduced by Gary Guth and Ciprian Manolescu, and inspired by work of Jiakai Li and others in Seiberg-Witten theory. After sketching their construction, we will describe an extension of the "hat" variant to 3-manifolds with boundary, and the algorithm this gives to compute it when the fixed set is connected. We will end with some open questions.

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.

Given a space X, one can study various "genera", which give cobordism invariants with interesting properties. In this talk, I will consider the case when X is the moduli space of quasimaps from a smooth projective curve C to a Nakajima quiver variety. I will present a number of results on the (twisted virtual equivariant) Hirzebruch genus and elliptic genus of such spaces. Such invariants are often determined by the case when C is genus zero. When the quiver variety is zero-dimensional, the quasimap moduli spaces generalize the variety parameterizing rank 0 quotients of a fixed vector bundle on C. In these cases, we can prove complete formulas which exhibit an a-postiori independence of the equivariant parameters, a phenomenon sometimes called "rigidity". This is based on work in progress with Reese Lance. 

TBD

The classical Livsic theorem says that a H\"older cocycle over a transitive Anosov diffeomorphism/flow is a coboundary if and only if it satisfies the periodic obstruction on all periodic orbits. It is natural to ask whether satisfying the periodic obstruction for all closed orbits of period at most $T$ is enough to conclude that the cocycle is, in some quantitative sense, close to being a coboundary. We show that for transitive Anosov flows, this is indeed enough to find an approximate solution to the cohomological equation with error decaying exponentially in $T$, improving on the polynomial rates obtained first by S. Katok for contact flows in dimension 3, and then later Gouëzel and Lefeuvre in higher dimensions. This is joint work with Jonathan DeWitt, Spencer Durham, and James Marshall Reber.

I would like to think that they can, and will illustrate by sharing some work my collaborators and I have done on the monkey visual system, which is very similar to that of humans. Specifically, I will focus on two visual properties: one is used in the detection of edges, the other is relevant when our eyes track moving objects. To explain the origin of these properties, simple mathematical ideas were first developed in idealized settings. They were then tested -- and fine-tuned -- via simulations using large-scale dynamical network models that are biologically more realistic.
 

The Birkhoff Ergodic Theorem describes typical behaviors and averaged quantities with respect to an invariant measure. In this talk, I will focus on "observable" events, equating observability with positive Lebesgue measure. From this observational viewpoint, "typical" means typical with respect to Lebesgue measure. This leads immediately to issues for attractors, where all invariant measures are singular. I will present highlights of developments in smooth ergodic theory that address these questions. The theory of physical and SRB measures applies to dynamical systems that are deterministic as well as random, in finite and infinite dimensions (where observability has to be interpreted differently). This body of ideas argue in favor of convergence of ergodic averages for typical orbits. But the picture is a little more complicated: In the last part of the talk, I will discuss some recent work that shows that in many natural settings (e.g. reaction networks), it is also typical for ergodic averages 
to fluctuate in perpetuity due to heteroclinic-like behavior.

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.