Georgia Institute of Technology College of Sciences

We discuss how chaos, i.e., sensitivity to initial conditions, arises in the setting of polygonal billiards. In particular, we give a complete classification of the rational polygons whose billiard flow is weak mixing in almost every direction, proving a longstanding conjecture of Gutkin. This is joint work with Jon Chaika and Giovanni Forni. No previous knowledge on the subject will be assumed.

A Dehn surgery slope p/q is said to be characterizing for a knot K if the homeomorphism type of the p/q-Dehn surgery along K determines the knot up to isotopy. I discuss advances towards a conjecture of McCoy that states that for any knot, all but at most finitely many non-integral slopes are characterizing.

The compressible Euler and Navier-Stokes equations describe the motion of compressible fluids. The defocusing nonlinear Schr\"odinger equation is a dispersive equation that has application in many physics areas. Through the Madelung transformation, the defocusing nonlinear Schr\"odinger equation is connected with the compressible Euler equation. In this colloquium I will start from the compressible Euler/Navier-Stokes equation and introduce the blow-up result called implosion. Then I will introduce the defocusing nonlinear Schr\"odinger equation and the longstanding open problem on the blow-up of its solutions in the energy supercritical regime. In the end I will talk about the Madelung transformation and its application to transfer the implosion from the compressible Euler to the defocusing nonlinear Schr\"odinger equation. During the talk I will mention our work with Gonzalo Cao-Labora, Javier Gómez-Serrano and Gigliola Staffilani on the first non-radial implosion result for those three equations.

(Note the unusual location!)

We study an extension of the k-Disjoint Paths Problem where, in addition to finding k disjoint paths joining k given pairs of vertices in a graph, we ask that those paths satisfy certain constraints expressable by abelian groups. We give an O(n^8) time algorithm to solve this problem under the assumption that the constraint can be expressed as avoiding a bounded number of group elements; moreover, our O(n^8) algorithm allows any bounded number of such constraints to be combined. Group-expressable constraints include, but not limited to: (1) paths of length r modulo m for any fixed r and m, (2) paths passing through any bounded number of prescribed sets of edges and/or vertices, and (3) paths that are long detours (paths of length at least r more than the distance between their ends for fixed r). The k=1 case with the modularity constraint solves problems of Arkin, Papadimitriou and Yannakakis from 1991. Our work also implies a polynomial time algorithm for testing the existence of a subgraph isomorphic to a subdivision of a fixed graph, where each path of the subdivision between branch vertices satisfies any combination of a bounded number of group-expressable constraints. In addition, our work implies similar results addressing edge-disjointness. It is joint work with Youngho Yoo.

We consider the mean field Bose gas on the unit torus at temperatures proportional to the critical temperature of the Bose—Einstein condensation phase transition. We discuss trace norm convergence of the Gibbs state to a state given by a convex combination of quasi-free states. Two consequences of this relation are precise asymptotic formulas for the two-point function and the distribution of the number of particles in the condensate. A crucial ingredient of the proof is a novel abstract correlation inequality. This is joint work with Nam Panh Tanh and Marcin Napiorkowski. 

In this talk, we will study random dynamical systems of smooth surface diffeomorphisms. Aaron Brown and Federico Rodriguez Hertz showed that, in this setting, hyperbolic stationary measures have the SRB property, except when certain obstructions occur. Here, the SRB property essentially means that the measure is absolutely continuous along certain “nice” curves (unstable manifolds). In this talk, we want to understand conditions that guarantee that SRB stationary measures are absolutely continuous with respect to the Lebesgue measure of the ambient space. Our approach is inspired on Tsujii’s “transversality” method, which he used to show Palis conjecture for partially hyperbolic endomorphisms. This is a joint work with Aaron Brown, Homin Lee and Yuping Ruan.