Georgia Institute of Technology College of Sciences

It has been known since the pioneering work of J.L. Lions in 1960s that 3D hyper-dissipative (HD) Navier-Stokes (NS) system exhibits global-in-time regularity as long as the hyper-diffusion exponent is greater or equal to 5/4.  One should note that at 5/4, the system is critical, i.e., the energy level and the scaling -invariant level coincide. What happens in the super-critical regime, the hyper-diffusion exponent being strictly between 1 and 5/4 remained a mystery. 

The goal of this talk is to demonstrate that as soon as the hyper-diffusion exponent is greater than 1, a class of monotone blow-up scenarios consistent with the analytic structure of the flow (prior to the possible singular time) can be ruled out (a sort of 'runaway train' scenario). The argument is in the spirit of the regularity theory of the 3D HD NS system in 'turbulent scenario' (in the super-critical regime) developed by Grujic and Xu, relying on 'dynamic interpolation' – however, it is much shorter, tailored to the class of blow-up profiles in view. This is a joint work with Aseel Farhat.

In this presentation, I will discuss my research in the field of data science, specifically in two areas: improving autoencoder interpolations and accelerating federated learning algorithms. My work combines advanced mathematical concepts with practical machine learning applications, contributing to both the theoretical and applied aspects of data science. The first part of my talk focuses on image sequence interpolation using autoencoders, which are essential tools in generative modeling. The focus is when there is only limited training data. By introducing a novel regularization term based on dynamic optimal transport to the loss function of autoencoder, my method can generate more robust and semantically coherent interpolation results. Additionally, the trained autoencoder can be used to generate barycenters. However, computation efficiency is a bottleneck of our method, and we are working on improving it. The second part of my presentation focuses on accelerating federated learning (FL) through the application of Anderson Acceleration. Our method achieves the same level of convergence performance as state-of-the-art second-order methods like GIANT by reweighting the local points and their gradients. However, our method only requires first-order information, making it a more practical and efficient choice for large-scale and complex training problems. Furthermore, our method is theoretically guaranteed to converge to the global minimizer with a linear rate.

Symmetry is prevalent in a variety of machine learning and scientific computing tasks, including computer vision and computational modeling of physical and engineering systems. Empirical studies have demonstrated that machine learning models designed to integrate the intrinsic symmetry of their tasks often exhibit substantially improved performance. Despite extensive theoretical and engineering advancements in symmetry-preserving machine learning, several critical questions remain unaddressed, presenting unique challenges and opportunities for applied mathematicians.

Firstly, real-world symmetries rarely manifest perfectly and are typically subject to various deformations. Therefore, a pivotal question arises: Can we effectively quantify and enhance the robustness of models to maintain an “approximate” symmetry, even under imperfect symmetry transformations? Secondly, although empirical evidence suggests that symmetry-preserving models require fewer training data to achieve equivalent accuracy, there is a need for more precise and rigorous quantification of this reduction in sample complexity attributable to symmetry preservation. Lastly, considering the non-convex nature of optimization in modern machine learning, can we ascertain whether algorithms like gradient descent can guide symmetry-preserving models to indeed converge to objectively better solutions compared to their generic counterparts, and if so, to what degree?

In this talk, I will provide an overview of my research addressing these intriguing questions. Surprisingly, the answers are not as straightforward as one might assume and, in some cases, are counterintuitive. My approach employs an interesting blend of applied probability, harmonic analysis, differential geometry, and optimization. However, specialized knowledge in these areas is not required. 

The Johnson filtration is a filtration of the mapping class group induced by the action of the mapping class group on the lower central series of the fundamental group of a surface.  A theorem of Johnson tells us that the first term of this filtration, called the Torelli group, is finitely generated for surfaces of genus at least 3.  We will explain work of Ershov—He and Church—Ershov—Putman, which uses Johnson's result to show that the kth term of the Johnson filtration is finitely generated for surfaces of genus g at least 2k - 1.  Time permitting, we will also discuss some extensions of these ideas.  In particular, we will explain how to show that the terms of the Johnson filtration are finitely presented assuming the Torelli group is finitely presented.