Georgia Institute of Technology College of Sciences

Sampling from the Gibbs distribution is a long-standing problem studied across various fields. Among many sampling algorithms, Langevin dynamics plays a crucial role, particularly for high-dimensional target distributions. In practical applications, accelerating sampling dynamics is always desirable. It has long been studied that adding an irreversible component to reversible dynamics, such as Langevin, can accelerate convergence. Concrete constructions of irreversible components have also been explored in specific scenarios. However, a general strategy for such construction is still elusive. In this talk, I will introduce the concept of leveraging irreversibility to accelerate general dynamics, along with the quantification of irreversible dynamics. Our theory is mainly based on designing a modified entropy functional originally developed for linear kinetic equations (Dolbeault et al., 2015).

In her thesis, Maryam Mirzakhani counted the number of simple closed geodesics of bounded length on a (real) hyperbolic surface. This breakthrough theorem and the subsequent explosion of related results use techniques and draw inspiration from Teichmüller theory, symplectic geometry, surface topology, and homogeneous dynamics. In this talk, I’ll discuss some of these connections and a qualitative strengthening of her theorem, describing what these curves, and their complements, actually (generically) look like. This is joint work with Francisco Arana-Herrera.

The homotopy continuation is a widely recognized method for finding solutions to polynomial systems by tracking the homotopy paths of solutions. However, the current implementation of homotopy continuation relies on heuristics, and hence it requires certification to verify its correctness. We discuss two modalities of certification in algebraic geometry exploiting interval arithmetic. The first is certified homotopy tracking using the Krawczyk method which guarantees correct tracking without path jumping. The second is Smale’s alpha theory over regions for faster certification. We discuss experimental results to demonstrate the effectiveness of these new methods. This talk is a preliminary report of two separate ongoing works.

A successful trend in modern extremal/probabilistic combinatorics is the investigation of how well classical theorems, like those of Ramsey, Turán, and Szemerédi, hold in sparse random contexts. Graph and hypergraph container methods have played a big role in improving our knowledge of these sparse structures. I will present joint work with Jozsef Balogh and Haoran Luo on a random version of the Erdős-Ko-Rado Theorem and Sperner's Theorem, giving the flavor of some graph container techniques.