Georgia Institute of Technology College of Sciences

Motivated by the Soliton Resolution Conjecture, the study of dynamics of multi-solitons has been crucial to understand the  long-time behavior of solutions for dispersive PDEs.

In this talk, we consider one-dimensional L2 supercritical nonlinear Schrödinger equation.

It is well-known that the solitons for this model are unstable, but conditional asymptotic stability for a single soliton was obtained in the pioneering work of Krieger and Schlag. In this talk, using the linear and scattering theory developed in our previous work, we show the conditional strong asymptotic stability for any multi-solitons with large separation in the speed. More precisely,  this solution of the supercritical NLS will converge asymptotically in the H1 norm to a finite of multi-solitons moving with constant speeds plus a radiation (Scattering of the remainder).  Finally, at the end of the talk, we discuss our ongoing research related to this topic.  This is a joint work with Gong Chen.

We study the problem of computing the convex hull of a set $S \subseteq \mathbb{R}^n$ defined by three quadratic inequalities. A simple way to generate inequalities valid on $S$ is to take nonnegative linear combinations, called aggregations, of the defining inequalities. We study the set defined by aggregations using topological duality results for quadratic inequalities. In the case of three quadratic inequalities, this relates aggregations to an algebraic curve. This viewpoint allows us to find new cases for which the convex hull of $S$ can be recovered by aggregations. Joint work with Greg Blekherman.

We rigorously prove the filamentation phenomenon for a class of weak solutions to the Euler equations known as vortex caps. Vortex caps are characteristic functions representing time-evolving sets of Lagrangian type, with energy preserved at all times. The filamentation of vortex caps is characterized by L^1 -stability alongside unbounded growth of the perimeter of their interfaces. We recall the existence and stability results for vortex caps on the sphere, based on Yudovich theory. Using L^1 -stability, we derive a lower bound for the growth of the perimeter of vortex caps over time.

Filamentous materials may exhibit structure-dependent material properties and function that depend on their entanglement. Even though intuitively entanglement is often understood in terms of knotting or linking, many of the filamentous systems in the natural world are not mathematical knots or links. In this talk, we will introduce a novel and general framework in knot theory that can characterize the complexity of open curves in 3-space. This leads to new metrics of entanglement of open curves in 3-space that generalize classical topological invariants, like for example, the Jones polynomial and Vassiliev invariants. For open curves, these are continuous functions of the curve coordinates and converge to topological invariants of classical knots and links when the endpoints of the curves tend to coincide. These methods provide an innovative approach to advance important questions in knot theory. As an example, we will see how the theory of linkoids enables the first, to our knowledge, parallel algorithm for computing the Jones polynomial.

Importantly, this approach opens exciting applications to systems that can be modeled as open curves in 3-space, such as polymers and proteins, for which new quantitative relationships between their structure and material properties become evident. As an example, we apply our methods to proteins to understand the interplay between their structures and functions. By analyzing almost all protein structures in the Protein Data Bank, we derive for the first time a quantitative representation of the topology/geometry of the Topological Landscape of proteins. We show that 3 topological and geometrical parameters alone can predict the biological classifications of proteins with high accuracy. Moreover, preliminary results show that our proposed topological metrics based on static protein structures alone correlate with protein dynamics and protein function. The methods and results represent a new framework for advancing knot theory, as well as its applications to filamentous materials, which can be validated by experimental data and integrated into machine-learning algorithms.