Georgia Institute of Technology College of Sciences

For integers $k>\ell\ge0$, a graph $G$ is $(k,\ell)$-stable if  $\alpha(G-S)\geq \alpha(G)-\ell$ for every    
$S\subseteq V(G)$ with $|S|=k$. A recent result of Dong and Wu [SIAM J.
Discrete Math. 36 (2022) 229--240] shows that every $(k,\ell)$-stable 
graph $G$  satisfies $\alpha(G) \le  \lfloor ({|V(G)|-k+1})/{2}\rfloor+\ell$.  A $(k,\ell)$-stable graph $G$   is   tight if $\alpha(G) = \lfloor ({|V(G)|-k+1})/{2}\rfloor+\ell$; and  $q$-tight for some integer $q\ge0$ if $\alpha(G) = \lfloor ({|V(G)|-k+1})/{2}\rfloor+\ell-q$.
In this talk, we first prove  that for all $k\geq 24$, the only tight $(k, 0)$-stable graphs are $K_{k+1}$ and  $K_{k+2}$, answering a question of Dong and Luo [arXiv: 2401.16639]. We then prove that  for all nonnegative integers $k, \ell, q$ with $k\geq 3\ell+3$, every $q$-tight $(k,\ell)$-stable graph has at most  $k-3\ell-3+2^{3(\ell+2q+4)^2}$ vertices, answering a question of Dong and Luo in the negative.   \\  

This is joint work with Xiaonan Liu and Zhiyu Wang. 

Diffusion models, particularly score-based generative models (SGMs), have emerged as powerful tools in diverse machine learning applications, spanning from computer vision to modern language processing. In the first part of this talk, we delve into the generalization theory of SGMs, exploring their capacity for learning high-dimensional distributions. Our analysis show that SGMs achieve a dimension-free generation error bound when applied to a class of sub-Gaussian distributions characterized by certain low-complexity structures.  In the second part of the talk, we consider the application of diffusion models in solving partial differential equations (PDEs). Specifically, we present the development of a physics-guided diffusion model designed for reconstructing high-fidelity solutions from their low-fidelity counterparts. This application showcases the adaptability of diffusion models and their potential to scientific computation.  

Atlanta Combinatorics Colloquium Hosted by Georgia Tech

Abstract: Many tools have been developed in combinatorics to study global structure in finite graphs. One such tool is called Szemerédi's regularity lemma, which gives a structural decomposition for any large finite graph. Beginning with work of Alon–Fischer–Newman, Lovász–Szegedy, and Malliaris–Shelah, it has been shown over the last 15 years that regularity lemmas can be used to detect structural dichotomies in graphs, and that these dichotomies have deep connections to model theory. In this talk, I present extensions of this type of result to arithmetic regularity lemmas, which are analogues of graph regularity lemmas, tailored to the study of combinatorial problems in finite groups. This work uncovered tight connections between tools from additive combinatorics, and ideas from the model theoretic study of infinite groups.