Georgia Institute of Technology College of Sciences

We consider classical scalar fields in dimension 1+1 with a
self-interaction potential being a symmetric double-well. Such a model
admits non-trivial static solutions called kinks and antikinks. A kink
cluster is a solution approaching, for large positive times, a
superposition of alternating kinks and antikinks whose velocities
converge to 0 and mutual distances grow to infinity. Our main result is
a determination of the asymptotic behaviour of any kink cluster at the
leading order.
Our results are partially inspired by the notion of "parabolic motions"
in the Newtonian n-body problem. I will present this analogy and mention
its limitations. I will also explain the role of kink clusters as
universal profiles for formation of multi-kink configurations.
This is a joint work with Andrew Lawrie.

This talk will be about the convergence problem in mean field control (MFC), i.e. the challenge of rigorously justifying the convergence of certain "symmetric" N-particle control problems towards their mean field counterparts. On the one hand, this convergence problem is already well-understood from a qualitative perspective, thanks to powerful probabilistic techniques based on compactness. On the other hand, quantitative results (i.e. rates of convergence) are more difficult to obtain, in large part because the value function of the mean field problem (which is also the solution to a certain Hamilton-Jacobi equation on the Wasserstein space) may fail to be C^1, even if all the data is smooth. After giving an overview of the convergence problem, I will discuss the results of two recent joint works with Cardaliaguet, Daudin, Delarue, and Souganidis, in which we use some ideas from the theory of viscosity solutions to overcome this lack of regularity and obtain rates of convergence of the N-particle value functions towards the value function of the corresponding MFC problem.

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.