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. 

Given an integer $k\ge1$, an edge-$k$-coloring of a graph $G$ is an assignment of $k$ colors $1,\ldots,k$ to the edges of $G$ such that no two adjacent edges receive the same color. A vertex-distinguishing (resp. sum-distinguishing) edge-$k$-coloring of $G$ is an edge-$k$-coloring such that for any two distinct vertices $u$ and $v$, the set (resp. sum) of colors taken from all the edges incident with $u$ is different from that taken from all the edges incident with $v$. The vertex-distinguishing chromatic index (resp. sum-distinguishing chromatic index), denoted $\chi'_{vd}(G)$ (resp.  $\chi'_{sd}(G)$), is the smallest value $k$ such that $G$ has a vertex-distinguishing edge-$k$-coloring (resp. sum-distinguishing edge-$k$-coloring). Let $G$ be a   $d$-regular graph on $n$ vertices, where $n$ is even and sufficiently large. We show that $\chi'_{vd}(G) =d+2$ if $d$ is arbitrarily close to $n/2$ from above, and $\chi'_{sd}(G) =d+2$ if $d\ge \frac{2n}{3}$. 

This is joint work with Yuping Gao and Guanghui Wang.

Given a set in some manifold M, what is the probability that a quantum particle freely traveling in M spends a positive amount of time in the given set? Sets for which there is a positive (uniform) probability after some long time will be called observation sets. We will survey the well-studied case of compact manifolds and discuss our recent extension of these results to the non-compact setting of the whole Euclidean space. This is joint work with Perry Kleinhenz.  

In his 1956 paper "On manifolds homeomorphic to the 7-sphere'', John Milnor constructed some examples of manifolds that are homeomorphic, but not diffeomorphic, to the standard unit sphere. They are now called exotic 7-spheres. This example established that the differential structure of a manifold can carry information not given by its topological structure. Thus, Milnor founded differential topology as a stand-alone field. On my first reading of the paper, I thought that many of the choices Milnor made on his road to constructing the first exotic spheres seemed rather strange and arbitrary. Why 7-spheres? Why look for them among $S^3$-bundles over $S^4$ with structure group $SO(4)$? And what's the motivation behind his complicated mod 7-valued lambda invariant that detects exotica in these examples? Fortunately, Milnor answered some of these questions in his essay "Classification of $(n-1)$-connected $2n$-dimensional manifolds and the discovery of exotic spheres''. This talk is an attempt to understand this essay. 

Let $f$ be an endomorphism of projective space defined over a number field.  When counting rational points ordered by a certain "canonical" height function attached to $f$, we encounter a mysterious asymptotic constant in the main term.  This constant is a product of local factors over the primes of bad reduction of $f$; and these local factors (which take the form of $v$-adic integrals) are rather difficult to calculate explicitly.  In this talk I will present my partial progress towards evaluating these integrals.  No knowledge of arithmetic dynamics will be assumed.

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.

A binary shuffle square is a binary word of even length that can be partitioned into two disjoint, identical subwords. While recognizing shuffle squares is NP-hard, we show that they are surprisingly ubiquitous. We prove that a uniformly random binary word $s$ of length $2n$ is a shuffle square with probability $\frac 12-o(n^{-1/15})$, verifying a conjecture of He, Huang, Nam, and Thaper. In particular, almost every binary word is at most two bit-deletions away from a shuffle square, giving the best possible average case for the “Longest Twin” problem.

By revealing the bits of $s$ sequentially,  we reformulate the problem as a discrete stochastic process. We track the evolution of a “buffer set”, a collection of suffixes produced by the revealed bits. In this setting, there is a simple greedy algorithm which behaves like a SSRW; we define a local optimization which creates a negative bias. We also show that the buffer set is robust enough to absorb small defects, yielding a perfect partition with high probability.

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.