Georgia Institute of Technology College of Sciences

There are two standard ways to associate a principally polarized abelian variety (ppav) to a smooth algebraic curve X of genus g. The Jacobian variety Jac(X) is a ppav of dimension g. An etale double cover X’->X determines the Prym variety Prym(X’/X), which is a ppav of dimension g-1. These two objects are related by Recillas’ trigonal construction: given an etale double cover X’->X of a trigonal curve X, we can construct a tetragonal curve Y such that Prym(X’/X) is isomorphic to Jac(Y).

I will talk about a tropical version of the trigonal construction, where algebraic curves are replaced by metric graphs and ppavs by real tori with integral structure. Given a double cover X’->X of a trigonal graph X, we obtain a tetragonal graph Y such that the tropical Prym variety Prym(X’/X) and the tropical Jacobian Jac(Y) are isomorphic.

This construction has two applications. First, we can use it to compute the second moment of the tropical Prym variety for g up to 4, and conjecturally for all g, which has arithmetic applications. Second, the tropical trigonal construction provides an explicit resolution of the Prym—Torelli map in genus 4.

TBD

Mean-field games (MFG) theory is a mathematical framework for studying large systems of agents who play differential games. In the PDE form, MFG reduces to a Hamilton-Jacobi equation coupled with a continuity or Kolmogorov-Fokker-Planck equation. Theoretical analysis and computational methods for these systems are challenging due to the absence of strong regularizing mechanisms and coupling between two nonlinear PDE.

One approach that proved successful from both theoretical and computational perspectives is the variational approach, which interprets the PDE system as KKT conditions for suitable convex energy. MFG systems that admit such representations are called potential systems and are closely related to the dynamic formulation of the optimal transportation problem due to Benamou-Brenier. Unfortunately, not all MFG systems are potential systems, limiting the scope of their applications.

I will present a new approach to tackle non-potential systems by providing a suitable interpretation of the Benamou-Brenier approach in terms of monotone inclusions. In particular, I will present advances on the discrete level and numerical analysis and discuss prospects for the PDE analysis.

Ryser (1951) provided the conditions under which any $r\times s$ Latin rectangle can be extended to an $n\times n$ Latin square. In this talk, we provide various generalizations of this result in higher dimensions. We also proof an analogue of Ryser’s theorem for symmetric latin cubes.

How densely can one pack spheres in $d$-dimensional space? It is not too hard to show a lower bound of $2^{-d}$. (The only known upper bounds are exponentially larger.) Various proofs of lower bounds of the form $cd2^{-d}$ have been given; recently, Campos, Jenssen, Michelen, and Sahasrabudhe gave the first asymptotic improvement on such bounds in 75 years. I will discuss an extension of this improvement to packing other shapes in high dimensions, along with some connections to log-concave probability.

Let $f,g$ be $C^2$ area-preserving Anosov diffeomorphisms on $\mathbb{T}^2$ which are topologically conjugated by a homeomorphism $h$. It was proved by de la Llave in 1992 that the conjugacy $h$ is automatically $C^{1+}$ if and only if the Jacobian periodic data of $f$ and $g$ are matched by $h$ for all periodic orbits. We prove that if the Jacobian periodic data of $f$ and $g$ are matched by $h$ for all points of some large period $N\in\mathbb{N}$, then $f$ and $g$ are ``approximately smoothly conjugate." That is, there exists a a $C^{1+\alpha}$ diffeomorphism $\overline{h}_N$ that is exponentially close to $h$ in the $C^0$ norm, and such that $f$ and $f_N:=\overline{h}_N^{-1}\circ g\circ \overline{h}_N$ is exponentially close to $f$ in the $C^1$ norm.

Zoom link - 

https://gatech.zoom.us/j/5506889191?pwd=jIjsRmRrKjUWYANogxZ2Jp1SYdaejU.1

Meeting ID: 550 688 9191

Passcode: 604975

The incipient infinite cluster was first proposed by physicists in the 1970s as a canonical example of a two-dimensional medium on which random walk is subdiffusive. It is the measure obtained in critical percolation by conditioning on the existence of an infinite cluster, which is a probability zero event. Kesten presented the first rigorous two-dimensional construction of this object as a weak limit of the one-arm event. In high dimensions, van der Hofstad and Jarai constructed the IIC as a weak limit of the two-point connection using the lace expansion. Our work presents a new high-dimensional construction which is "robust", establishing that the weak limit is independent of the choice of conditioning. The main tools used are Kesten's original two-dimensional construction combined with Kozma and Nachmias' regularity method. Our robustness allows for several applications, such as the explicit computation of the limiting distribution of the chemical distance, which forms the content of our upcoming project. This is joint work with Shirshendu Chatterjee, Jack Hanson, and Philippe Sosoe. The preprint can be found at https://arxiv.org/abs/2502.10882.

Using the theory of total linear stability for Dynkin quivers and an interplay between the Bruhat order and the noncrossing partition lattice, we define a family of triangulations of the permutohedron indexed by Coxeter elements.  Each triangulation is constructed to give an explicit homotopy between two complexes (the Salvetti complex and the Bessis--Brady--Watt complex) associated to two different presentations of the corresponding braid group (the standard Artin presentation and Bessis's dual presentation).  Our triangulations have several notable combinatorial properties. In addition, they refine similar Bruhat interval polytope decompositions of Knutson, Sanchez, and Sherman-Bennett. This is based on joint work with Melissa Sherman-Bennett and Nathan Williams.