Georgia Institute of Technology College of Sciences

For an integer $r\geq 2$, the $K_{r}$-free chromatic number of a graph $G$, denoted by $\chi_{r}(G)$, is the minimum size of a partition of the set of vertices of $G$ into parts each of which induces a $K_{r}$-free graph. In this setting, the $K_{2}$-free chromatic number is the usual chromatic number.

Which are the unavoidable induced subgraphs of graphs of large $K_{r}$-free chromatic number? Generalizing the notion of $\chi$-boundedness, we say that a hereditary class of graphs is $\chi_{r}$-bounded if there exists a function which provides an upper bound for the $K_{r}$-free chromatic number of each graph of the class in terms of the graph's clique number. 

With an emphasis on a generalization of the Gy\'arf\'as-Sumner conjecture for $\chi_{r}$-bounded classes of graphs and on polynomial $\chi$-boundedness, I will discuss some recent developments on $\chi_{r}$-boundedness and related open problems. 

Based on joint work with Mathieu Rundstr\"om and Sophie Spirkl, and with Bartosz Walczak.
 

In this talk, I will discuss our recent theoretical advancements in generative modeling. The first part of the presentation will focus on learning distributions with symmetry. I will introduce results on the sample complexity of empirical estimations of probability divergences for group-invariant distributions, and present performance guarantees for GANs and score-based generative models that incorporate symmetry. Notably, I will offer the first quantitative comparison between data augmentation and directly embedding symmetry into models, highlighting the latter as a more fundamental approach for efficient learning. These findings underscore how incorporating symmetry into generative models can significantly enhance learning efficiency, particularly in data-limited scenarios. The second part will cover $\alpha$-divergences with Wasserstein-1 regularization. These divergences can be interpreted as $\alpha$-divergences constrained to Lipschitz test functions in their variational form. I will demonstrate how generative learning can be made agnostic to assumptions about target distributions, including those with heavy tails or low-dimensional and fractal supports, through the use of these divergences as objective functionals. I will outline the conditions for the finiteness of these divergences under minimal assumptions on the target distribution along with the variational derivatives and gradient flow formulation associated with them. This framework provides guarantees for various machine learning algorithms that optimize over this class of divergences. 

In this talk, for an ergodic probability preserving system $(X,\mathcal{B},m,T)$, we will discuss the existence of a function $f:X\to \mathbb{Z}^d$, whose corresponding cocycle satisfies the $d$-dimensional local central limit theorem.
As an application, we resolve a question of Huang, Shao and Ye, and Franzikinakis and Host regarding non-convergence in $L^2$ of polynomial multiple averages of non-commuting zero entropy transformations. If time allows, we will also discuss the first examples of failure of multiple recurrence for zero entropy transformations along polynomial iterates. This is joint work with Zemer Kosloff (arXiv:2409.05087). 

The totally non-negative Grassmannian is the set of points in a real Grassmannian such that all Plucker coordinates have the same sign (some can be zero). I will show how points in totally non-negative Grassmannians arise from the spaces of polynomials in one variable whose Wronskian has only real roots. Then I will discuss a similar result for the spaces of quasi-exponentials.

The main statements of this talk should be understandable to an undergraduate student. Somewhat surprisingly, the proofs use the theory of quantum integrable systems related to $GL(n)$. I will try to explain the logic of such proofs in a gentle way.

This talk is based on a joint work with S. Karp and V. Tarasov.