Georgia Institute of Technology College of Sciences

A matroid is modular if its lattice of flats is self-dual. Prototypical examples include the Boolean matroid $U_{n,n}$ and finite projective geometry matroid $M(\mathbb{F}_q^n)$. We discovered that the space of quotients of any modular matroids has remarkable structure. For instance, quotients of the Boolean matroid $U_{n,n}$, also known as the set of all matroids on $[n]$, are equipped with operations such as matroid duality and matroid intersection. All these operations exist canonically for quotients of arbitrary modular matroids. We will showcase some implications of these operations, including a generalization of stable intersection on the Bergman fan of any modular matroid.

We discuss a new problem by Dmitri Burago and Anton Petrunin. Consider a $C^2$-smooth surface embedded in $\RR^3$ with principal curvatures $\leq 1$. Does such a surface necessarily enclose a volume at least that of the unit ball? We address results from two opposite perspectives: On the one hand, we can construct a counterexample by deforming Lagunov's fishbowl (a fattened Bing's house); on the other hand, there are positive results with additional conditions. Many problems remain open.

Finding a path between two probability distributions is a recurring problem in optimal transport, generative modeling, population dynamics, and control. A broad class of these problems can be formulated through an action functional defined over curves of probability distributions. The corresponding first-order optimality conditions lead to a coupled Hamilton–Jacobi and continuity system, which can be written as a boundary-value Hamiltonian flow in density space.

Computing these paths from samples becomes challenging in moderate and high dimensions. In this talk, I will discuss three approaches for learning and optimizing them: Parametric Density Path Optimization (PDPO), Hamiltonian Rectification (HR), and Neural Multiple Shooting (NMS) with a focus on NMS. 

The central idea of NMS is to reformulate the boundary-value problem in density space as a multiple-shooting problem in particle space. Since the particle dynamics are determined by the prescribed Hamiltonian system once the momentum is specified, the unknowns reduce to momentum maps that initialize a sequence of Hamiltonian initial-value problems. The resulting hybrid method combines neural approximation with classical multiple shooting: neural networks determine the momentum at the beginning of each shooting segment, while an ODE solver propagates the corresponding trajectories according to the prescribed dynamics.

I will illustrate the method through examples involving obstacle avoidance, interacting particle systems, double-integrator dynamics with smooth drag, and unicycle dynamics with obstacles and mean-field interactions.

(first of two talks; the second is on Sep. 29)

A random geometric graph (RGG) is generated by first sampling $n$ latent points independently and uniformly from the unit sphere in $R^d$, and then connecting each pair of points if their inner product exceeds a threshold. We study the sharp detection threshold---the largest dimension at which the RGG can be statistically distinguished from the Erdős--Rényi graph with the same edge density $p$. This threshold is conjectured to be $d \asymp (n h(p))^3$, where $h(p)$ is the binary entropy function. Previous works proved this conjecture for dense graphs with constant $p$ and, up to polylogarithmic factors, very sparse graphs with constant average degrees. In this series of two talks, I will discuss a resolution of this conjecture. This is based on joint work with Hang Du, Nike Sun, Yihong Wu, and Jiaming Xu.

On a given Riemann surface, we may talk about various projective structures that are compatible with the given complex structure. These structures are equivalent to what are known as $sl_2$ opers, or in terms of Faltings’ paper that we mainly follow, as permissible connections. Then there are several interesting questions we may ask about the monodromies of these objects, and the one we focus on today is whether the monodromy group is conjugate into $PSL_2(\mathbb R)$ in $PSL_2(\mathbb C)$ (“the real monodromy property”). In this talk, we illustrate (literally!) Faltings’ idea on this matter by some pictures and examples, and explain how uniformization and accessory problem fit into this theory. If time permitted, we will also see how this shows up in the recent Analytic Langlands program proposed by Etingof, Frenkel and Kazhdan.

Working productively with modern AI tools requires looking beyond the products as they are currently packaged, identifying the underlying capabilities that are useful for our own goals, and then building systems to make those capabilities an honest net benefit in practice. The aim is to make the tools suit our interests while preserving what we value in mathematics. I will discuss this perspective through examples centered on research and verification.

We develop an extension of the Stein-Malliavin calculus which allows to measure the Wasserstein distance between the probability distributions of $ (X, Y)$ and $(Z,Y)$, where $X,Y$ are arbitrary random vectors and $ Z\sim N(0, \sigma ^{2})$ is independent of $Y$. In particular, this method allows to quantify the asymptotic independence between sequences of random variables and vectors. We will discuss some particular applications of this method to various limit theorems.

The multipermutohedral Chow ring was introduced in a series of papers by Clader, Damiolini, Eur, Huang, Li, and Ramadas to study moduli spaces with cyclic symmetry. It generalizes Chow rings of permutohedral varieties and type-B Coxeter arrangements. In this talk, we establish the combinatorial structure of the multipermutohedral Chow ring through an explicit Gröbner basis, yielding a Feichtner-Yuzvinsky-type monomial basis and a formula for the Hilbert series. Using this formula, we refine the palindromicity of the Hilbert series. From a representation-theoretic perspective, we also compute the equivariant Hilbert series under two natural group actions and construct combinatorial maps that establish equivariant unimodality and palindromicity.