In this talk, we first discuss existence and uniqueness of weak solutions to general time fractional equations and give their probabilistic representation. We then talk about sharp two-  sided estimates for fundamental solutions of general time fractional equations in metric measure spaces. This is a joint work with  Zhen-Qing Chen(University of Washington, USA), Takashi Kumagai (RIMS, Kyoto University, Japan) and Jian Wang (Fujian Normal University, China).

In this talk, I will discuss recent results produced with co-authors Ivan Blank (KSU) and Brian Benson (UCR) regarding a formulation of the Mean Value Theorem for the Laplace-Beltrami operator on smooth Riemannian manifolds. We define the sets upon which mean values of (sub)-harmonic functions are computed via a particular obstacle problem in geodesic balls. I will thus begin by discussing the classical obstacle problem and then an intrinsic formulation on manifolds developed in our recent paper. After demonstrating how the theory of obstacle problems is leveraged to produce our Mean Value Theorem, I will discuss local and global theory for our family of mean value sets and potential connections between the properties of these sets and the geometry of the underlying manifold.

A reaction-diffusion initial-boundary problem with a Caputo time derivative of order $\alpha\in (0,1)$ is considered. The solution of such a problem is discussed; it is shown that in general the solution has a weak singularity near the initial time $t=0$, and sharp pointwise bounds on the derivatives of this solution are derived. These bounds are used in a new analysis of the standard L1 finite difference method for the time derivative combined with a standard finite difference approximation for the spatial derivative. This analysis encompasses both uniform meshes and meshes that are graded in time, and includes new stability and consistency bounds. The final convergence result shows clearly how the regularity of the solution and the grading of the mesh affect the order of convergence of the difference scheme, so one can choose an optimal mesh grading to solve the problem numerically.

We study the  XXZ quantum spin chain in  a random field. This model is particle number preserving, which allows  the reduction to an infinite system of discrete many-body random Schrodinger operators.  We exploit this reduction to prove a form of  Anderson localization in the droplet  spectrum of the XXZ quantum spin chain Hamiltonian. This yields a strong form of dynamical exponential clustering for eigenstates  in the droplet spectrum: For any pair of local observables,  the sum of the associated correlators over these states decays exponentially  in the distance between the  local observables. Moreover,  this exponential clustering persists under the time evolution in the  droplet spectrum.

We will discuss certain new directions in the nexus of ideas that originate in Optimal Mass Transport (OMT) and the Schroedinger Bridge Problem (SBP). We will begin with a brief historic overview, explain the relation between OMT and SBP, discuss applications in control, physics, and networks,  and we will conclude with generalizations to the setting of matrix-valued and vector-valued distributions. This final chapter is pertinent to quantum mechanics as it explains the Lindblad equation of open quantum systems as gradient flow of the von Neumann entropy, and it is pertinent to multivariable signal and image processing (DTI, color, etc.). 

The talk is based on joint work with Yongxin Chen (ISU), Michele Pavon (University of Padova), and Allen Tannenbaum (Stony Brook).