This lecture is intended to be a survey of results of the type in the title. Historically, these ideas seem to have come to a head with the work of N. Varopoulous in the mid 1980’s culminating with the very nice little book: Geometry and Analysis on Groups, by Varopoulous, Saloff-Coste, Coulhon Cambridge U. Press, 1993. Other more recent work includes that of Saloff-Coste, Hebey, and others. Also, I will try to do this without dwelling on the definition/properties of Riemannian manifolds, but mention some of the major theorems about such that we need to formulate the results. Also, I’ll try to discuss some nonlinear PDE on smooth R-manifolds as time permits.

We show how the fractional moment method of Aizenman and Molchanov can be applied to a class of Anderson-type models with non-monotone potentials, to prove (spectral and dynamical) localization. The main new feature of our argument is that it does not assume any a priori Wegner-type estimate: the (nearly optimal) regularity of the density of states is established as a byproduct of the proof. The argument is applicable to finite-range alloy-type models and to a class of operators with matrix-valued potentials.

A nonlinear scattering transform is studied for the two-dimensional Schrodinger equation at zero energy with a radial potential. For a class of potentials call "conductivity type", it is known that there are no singularities in this scattering transform. We will look at a family of perturbations of conductivity type potentials and show where the singularities in their scattering transforms occur. This is some of the first work explicitly calculating the behavior of these singularities.

I will review recent results on singularity formation under the mean-curvature flow. In particular, l will discuss the phenomena of neck-pinching and collapse of closed surfaces.

The impact of charged-particles on the human’s life is constantly increasing, due to their importance in such domains as plasma physics, industrial processes, biology etc. It is related to a large variety of physical situations and has complex multiscale character. In this talk, I will explore the mathematical theory for the charged particles including the kinetic equations and continuum field models. In particular, I will discuss the diffusion limit of Vlasov-Poisson-Fokker-Planck (VPFP) equations to the Poisson-Nernst-Planck (PNP) equations for multispecies charged particles, which are widely used to describe the drift-diffusion of electrons and holes in semiconductors, as well as the movement of ions in solutions and protein channels. Besides, I will discuss the well-posedness and long-time behavior of the PNP equations with a nonlinear generation-recombination rate.

We will introduce the rudiments of a new theory of non-smooth solutions which applies to fully nonlinear PDE systems and extends Viscosity Solutions of Crandall-Ishii-Lions to the general vector case. Key ingredient is the discovery of a notion of Extremum for maps which extends min-max uniquely and allows for “nonlinear passage of derivatives” to test maps. The notions supports uniqueness, existence and stability results, preserving most features of the scalar viscosity counterpart. We will also discuss applications in vector-valued Calculus of Variations in L^\infty and Hamilton-Jacobi PDE with vector solution.

In this talk, I will review briefly the general Ericksen-Leslie system modeling the hydrodynamic motion of the nematic liquid crystals proposed by Ericksen and Leslie back in 1960’s. I will focus on the mathematical analysis of a simplified version of the Ericksen-Leslie system, proposed by Lin, which is a strong coupling between the Navier Stokes equation and the transported heat flow of harmonic maps into the two sphere. I will then present some recent results on the global existence of Leray-Hopf type weak solutions in dimension two, and several well-posedness results for small initial data in various function spaces in dimension three. It is based on joint works with Fanghua Lin, Junyu Lin, Tao Huang, and Jay Hineman.

In this talk I will describe some of my recent work on the resolvent estimates in L^p for the Stokes operator in Lipschitz domains. The results, in particular, imply that the Stokes operator in a three-dimensional Lipschitz domain generates a bounded analytic semigroup in L^p for p between 3 and 3/2. This gives an altermative answer to a conjecture of Michael Taylor.