Georgia Institute of Technology College of Sciences

A fully nonlinear surface dynamics of the time dependent potential flow of ideal incompressible fluid with a free surface is considered in two dimensional geometry. Arbitrary large surface waves can be efficiently characterized through a time-dependent conformal mapping of a fluid domain into the lower complex half-plane. We reformulate the exact Eulerian dynamics through a non-canonical nonlocal Hamiltonian system for the pair of new conformal variables. We also consider a generalized hydrodynamics for two components of superfluid Helium which has the same non-canonical Hamiltonian structure. In both cases the fluid dynamics is fully characterized by the complex singularities in the upper complex half-plane of the conformal map and the complex velocity. Analytical continuation through the branch cuts generically results in the Riemann surface with infinite number of sheets including Stokes wave, An infinite family of solutions with moving poles are found on the Riemann surface. Residues of poles are the constants of motion. These constants commute with each other in the sense of underlying non-canonical Hamiltonian dynamics which provides an argument in support of the conjecture of complete Hamiltonian integrability of surface dynamics. If we consider initial conditions with short branch cuts then fluid dynamics is reduced to the complex Hopf equation for the complex velocity coupled with the complex transport equation for the conformal mapping. These equations are fully integrable by characteristics producing the infinite family of solutions, including the pairs of moving square root branch points. The solutions are compared with the simulations of the full Eulerian dynamics giving excellent agreement.

In the past decade, deep learning has made astonishing breakthroughs in various real-world applications. It is a common belief that deep neural networks are good at learning various geometric structures hidden in data sets, such as rich local regularities, global symmetries, or repetitive patterns. One of the central interests in deep learning theory is to understand why deep neural networks are successful, and how they utilize low-dimensional data structures. In this talk, I will present some statistical learning theory of deep neural networks where data exhibit low-dimensional structures, such as lying on a low-dimensional manifold. The learning tasks include regression, classification, feature representation and operator learning. When data are sampled on a low-dimensional manifold, the sample complexity crucially depends on the intrinsic dimension of the manifold instead of the ambient dimension of the data. These results demonstrate that deep neural networks are adaptive to low-dimensional geometric structures of data sets.

The fractal uncertainty principle (FUP) roughly says that a function and its Fourier transform cannot both be concentrated on a fractal set. These were introduced to harmonic analysis in order to prove new results in quantum chaos: if eigenfunctions on hyperbolic manifolds concentrated in unexpected ways, that would contradict the FUP. Bourgain and Dyatlov proved FUP over the real numbers, and in this talk I will discuss an extension to higher dimensions. The bulk of the work is constructing certain plurisubharmonic functions on C^n. 

In this talk I will present some recent results in collaboration with Jessica Trespalacios where we consider Einstein-Belinski-Zakharov spacetimes and prove local and global existence, long time behavior of possibly large solutions and some applications to gravisolitons of Kasner type.