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.

A neural code C on n neurons is a collection of subsets of {1,2,...,n} which is used to encode the intersections of subsets U_1, U_2,...,U_n of some topological space. The study of neural codes reveals the ways in which geometric or topological properties can be encoded combinatorially. A prominent example is the property of max-intersection completeness: if a code C contains every possible intersection of its maximal codewords, then one can always find a collection of open convex U_1, U_2,..., U_n for which C is the code. In this talk I will answer a question posed by Curto et al. (2018), which asks if there is a way of determining max-intersection completeness from examination of the neural ideal, an algebraic counterpart to the neural code.

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 existence of global solutions for the Schrödinger equation
 i\partial_t u + \Delta u = P_d(u),
with nonlinearity $P_d$ homogeneous of degree $d$, has been extensively studied. Most results focus on the case with gauge invariant nonlinearity, where the solution satisfies several conservation laws. However, the problem becomes more complicated as we consider a general nonlinearity $P_d$. So far, global well-posedness for small data is known for $d$ strictly greater than the Strauss exponent. In dimension $3$, this Strauss exponent is $2$, making NLS with quadratic nonlinearity an interesting topic.

In this talk, I will present a result that shows the global existence and scattering for systems of quadratic NLS for small, localized data. To tackle the challenge presented by the $u\Bar{u}$-type nonlinearity, we require an $\epsilon$ regularization for the terms of this type in the system.