Events for 01/14/2025 from all calendars

Combinatorial Algebraic Geometry

Time: 12:45PM - 1:15PM

Location: Bloc 302

Speaker: Ruzho Sagayaraj, Texas A&M University

Title: Chebyshev Varieties

Abstract: Chebyshev polynomials are useful in function approximation as the root finding problem is better-conditioned in the basis of Chebyshev polynomials than in the familiar monomial basis. In this talk, I will introduce multivariate generalizations of Chebyshev polynomials and use them to define Chebyshev varieties parametrized by Chebyshev polynomials analogous to toric varieties parametrized by monomials. I will also discuss the geometry of Chebyshev varieties and list some applications. This talk is based on the paper Chebyshev varieties by Z. Bel-Afia, C. Meroni and S. Telen ( arXiv:2401.12140 ).

Following this presentation, there will be a more general discussion of arithmetic toric varieties.

Departmental Colloquia

Time: 4:00PM - 5:00PM

Location: Bloc 117

Speaker: Ye He

Title: Beyond Log-Concavity: Sampling Challenges and Advances in Multimodal and Heavy-Tailed Distributions

Abstract: Sampling from non-log-concave distributions poses significant challenges in a variety of fields, from Bayesian inference to computational physics and machine learning. Unlike log-concave distributions, which offer theoretical guarantees for efficient sampling, non-log-concave distributions often feature complex landscapes, including multimodality and heavy tails, that hinder standard algorithms from exploring the state space effectively. In this talk, I will discuss key obstacles and recent advances in sampling algorithms for non-log-concave distributions. First, I will explore the behavior of classical methods, such as Langevin Monte Carlo (LMC) and Proximal Sampler in the presence of multiple modes and heavy-tailed behaviors, highlighting issues like metastability and slow mixing. I will then introduce techniques designed to overcome these challenges, including using denoising diffusion and novel modifications to the Gaussian noise. This presentation aims to shed light on how these innovations bridge the gap between theory and practice, offering a more nuanced understanding of sampling in complex, high-dimensional spaces. By addressing these fundamental challenges, we can deepen our insight into the behavior of advanced sampling algorithms in non-log-concave regimes.