Georgia Institute of Technology College of Sciences

The compressible Euler and Navier-Stokes equations describe the motion of compressible fluids. The defocusing nonlinear Schr\"odinger equation is a dispersive equation that has application in many physics areas. Through the Madelung transformation, the defocusing nonlinear Schr\"odinger equation is connected with the compressible Euler equation. In this colloquium I will start from the compressible Euler/Navier-Stokes equation and introduce the blow-up result called implosion. Then I will introduce the defocusing nonlinear Schr\"odinger equation and the longstanding open problem on the blow-up of its solutions in the energy supercritical regime. In the end I will talk about the Madelung transformation and its application to transfer the implosion from the compressible Euler to the defocusing nonlinear Schr\"odinger equation. During the talk I will mention our work with Gonzalo Cao-Labora, Javier Gómez-Serrano and Gigliola Staffilani on the first non-radial implosion result for those three equations.

A Dehn surgery slope p/q is said to be characterizing for a knot K if the homeomorphism type of the p/q-Dehn surgery along K determines the knot up to isotopy. I discuss advances towards a conjecture of McCoy that states that for any knot, all but at most finitely many non-integral slopes are characterizing.

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.