Georgia Institute of Technology College of Sciences

We present in this talk some results concerning the metric measure spaces with lower Ricci curvature bounds. 

Firstly, we extend the technique of smoothing Riemannian metric by heat kernel pull back metrics to non-compact setting, and use it to solve a conjecture of De Philippis-Gigli. This is joint work with Brena-Gigli-Honda. Secondly, we study the second term in the short time expansion of the heat kernel pull back metrics and the connection with non-collapsed spaces. This is joint work with Honda. Finally, we use the 1D localization technique to extend some convexity results on the regular set and in the interior of such metric measure spaces.

Link: https://gatech.zoom.us/j/5491403383?pwd=Um1NM05MeWJMRnNuVHViQ1NWdHFaZz09 

Abstract: 

In this talk, we mainly focus on the applications of optimal transport theory from the following two aspects:

(1)Based on the theory of Wasserstein gradient flows, we develop and analyze a numerical method proposed for solving high-dimensional Fokker-Planck equations (FPE). The gradient flow structure of FPE allows us to derive a finite-dimensional ODE by projecting the dynamics of FPE onto a finite-dimensional parameter space whose parameters are inherited from certain generative model such as normalizing flow. We design a bi-level minimization scheme for time discretization of the proposed ODE. Such algorithm is sampling-based, which can readily handle computations in high-dimensional space. Moreover, we establish theoretical bounds for the asymptotic convergence analysis as well as the error analysis for our proposed method.

(2)Inspired by the theory of Wasserstein Hamiltonian flow, we present a novel definition of stochastic Hamiltonian process on graphs as certain kinds of inhomogeneous Markov process. Such definition is motivated by lifting to the probability space of the graph and considering the Hamiltonian dynamics on this probability space. We demonstrate some examples of the stochastic Hamiltonian process in classical discrete problems, such as the optimal transport problems and Schrödinger bridge problems (SBP).

The Bluejeans link: https://bluejeans.com/982835213/2740

We consider the problem of following quasi-periodic tori in perturbations of Hamiltonian systems which involve friction and external forcing.
In the first part, we study a family of dissipative standard maps of the cylinder for which the dissipation is a function of a small complex parameter of perturbation, $\varepsilon$.  We compute perturbative expansions formally in $\varepsilon$ and use them to estimate the shape of the domains of analyticity of invariant circles as functions of $\varepsilon$. We also give evidence that the functions might belong to a Gevrey class. The numerical computations we perform support conjectures on the shape of the domains of analyticity.

In the second part, we study rigorously the(divergent) series of formal expansions of the torus obtained using Lindstedt method.   We show that, for some systems in the literature, the series is Gevrey. We hope that the method of proof can be of independent interest: We develop KAM estimates for the divergent series. In contrast with the regular KAM method, we loose control of all the domains, so that there is no convergence, but we can generate enough control to show that the series is Gevrey.

https://bluejeans.com/417759047/0103

This thesis consists of four works in dynamical systems with a focus on billiards. In the first part, we consider open dynamical systems, where there exists at least a ``hole" of positive measure in the phase space which some portion of points in phase space escapes through that hole at each iterate of the dynamical system map. Here, we study the escape rate (a quantity that presents at what rate points in phase space escape through the hole) and various estimations of the escape rate of an open dynamical system. We uncover a reason why the escape rate is faster than expected, which is the convexity of the function defining escape rate. Moreover, exact computations of escape rate and its estimations are present for the skewed tent map and Arnold’s cat map.

In the second part of the thesis, we study physical billiards where the moving particle has a finite nonzero size. In contrast to mathematical billiards where a trajectory is excluded when it hits a corner point of the boundary, in physical billiards reflection of the physical particle (a ball) off a visible corner point is well-defined. Initially, we study properties of such reflections in a physical billiards. Our results confirm that the reflection considered in the literature about physical billiards are indeed no-slip friction-free (elastic) collisions.

In the third part of the thesis, we study physical Ehrenfests' wind-tree models, where we show that physical wind-tree models are dynamically richer than the well-known Lorentz gas model. More precisely, when we replace the point particle by a physical one (a ball), the wind-tree models show a new superdiffusive regimes that never been observed in any other model such as Lorentz gas.

Finally, we prove that typical physical polygonal billiard is hyperbolic at least on a subset of positive measure and therefore has a positive Kolmogorov-Sinai entropy for any positive radius of the moving particle.