Georgia Institute of Technology College of Sciences

Let $A = (a_{ij})$ be an $n \times n$ matrix with entries in a field $\mathbb{F}$ and let $f$ be a function defined on $\mathbb{F}$. The function naturally induces an entrywise transformation of $A$ via $f[A] := (f(a_{ij}))$. The study of such entrywise transforms that preserve various forms of matrix positivity has a rich and long history since the seminal work of Schoenberg. In this talk, I will discuss recent developments in the setting that the underlying field $\mathbb{F}$ is the real field, the complex field, and finite fields. I will also highlight some interesting connections between these problems with arithmetic combinatorics, finite geometry, and graph theory. Joint work with Dominique Guillot, Himanshu Gupta, and Prateek Kumar Vishwakarma.

In a highly influential paper from 1978, Lovász used topological methods to determine the chromatic number of the Kneser graph of the set of k-element subsets of a set with n elements. In this talk, we will discuss the Kneser graph of the set of triangulations of a convex n-gon and a recent proof that the chromatic number of this graph is n-2. The geometry of the associahedron will play a particularly important role in the argument. Based on a joint work with Anton Molnar, Cosmin Pohoata and Daniel Zhu.

Abstract:
This talk explores the Uzawa method, tracing its development from early applications in partial differential equations (PDEs) to modern advancements in optimization, image processing, and scientific computing. We will examine recent refinements for developing GPU-adaptive solvers for huge-scale linear programming and its extension to semidefinite programming arising in quantum information science. The discussion will also highlight the method's integration with deep learning and unrolling techniques for optimal control problems of PDEs, as well as its applications in industry.

Bio:

Xiaoming Yuan is a Professor in the Department of Mathematics at The University of Hong Kong. His research spans optimization, optimal control, scientific machine computing, and artificial intelligence. He is well recognized for his fundamental contributions to first-order optimization algorithms, including the Alternating Direction Method of Multipliers (ADMM), primal-dual methods, and proximal point algorithms. He also collaborates extensively with the AI and cloud computing industries. He led the development of the first automatic bandwidth allocation system for the cloud computing sector. His team was honored as a Franz Edelman Award Finalist in 2023.

Let us discuss how to use generative AI to help with math and coding.

My presentation features two scenarios:

Coding in LaTeX. Suppose you have a raw draft of what potentially could be a math paper. We will consider and apply simple AI tools that may help realizing the potential.

Coding in CAS. Suppose you have a raw idea for a package in a Computer Algebra System; your raw idea may be limited to a rough description of the input/output of a method you would like to implement. How far can an AI assistant take you? Can it autonomously code a working software package?