Georgia Institute of Technology College of Sciences

Many data in real-world applications are in a high-dimensional space but exhibit low-dimensional structures. In mathematics, these data can be modeled as random samples on a low-dimensional manifold. I will talk about machine learning tasks like regression and classification, as well as PDE simulations. We consider deep learning as a tool to solve these problems. 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. Our results demonstrate that deep neural networks can utilize low-dimensional geometric structures of data in machine learning and PDE simulations.

Due to linear superposition, solutions of a Linear Schrodinger Equation with a trapping potential,  produce a discrete  quasiperiodic part. When  a nonlinear perturbation is turned on,  it is known in principle, and proved in various situations,  that at small energies there is a phenomenon of standing wave selection where, up to radiation,  quasiperiodicity breaks down and there is convergence to a periodic wave.  We will discuss  this phenomenon in 1 D, where cubic nonlinearities are long range perturbations of the linear equations. Our aim is to show that a very effective framework to see these phenomena is provided by   a combination of the dispersion theory of  Kowalczyk, Martel and Munoz  along with  Maeda's  notion of Refined Profile.

I discuss some recent results, obtained jointly with David Wallauch, on the stability of self-similar wave maps under minimal regularity assumptions on the perturbation. More precisely, we prove the asymptotic stability of an explicitly known self-similar wave map in corotational symmetry. The key tool are Strichartz estimates for the linearized equation in similarity coordinates.

In this talk, I will present a method to construct nontrivial global solutions to some quasilinear wave equations in three space dimensions. Starting from a global solution to the geometric reduced system satisfying several pointwise estimates, we find a matching exact global solution to the original quasilinear wave equations. As an application of this method, we will construct nontrivial global solutions to Fritz John's counterexample $\Box u=u_tu_{tt}$ and the 3D compressible Euler equations without vorticity for $t\geq 0$.