The role of a biological membrane is to act as a barrier between ionic solutions. One way life controls ionic solution is through ion channels. A second more drastic way is by introducing a hole in the membrane itself. For example, in hemolysis, the osmotic swelling and rupture of a red-blood cell, a single hole forms in the membrane leading to the leak out of the contents of the cell. Similarly, in exocytosis a hole is formed by joining two membrane bilayers. These processes are mathematically challenging to study because they involve physical forces in the bulk and on surfaces with varying topology and predicting the time course is more consequential than the equilibrium end states. This talk will show how such complicated fluid mechanical problems yield to quantitative modeling and simulation when using the diffusive interface and energetic variational approach.  This is joint work with Fredric Cohen and Robert Eisenberg at the Rush University Medical Center in Chicago, IL

In this talk, I will first give a brief survey on Harnack inequalities for solutions of second-order elliptic and parabolic equations. Then I will describe my contribution on Harnack inequalities for non-divergent elliptic and parabolic equations on non-compact Riemannian manifolds.

It is well-known that the p-Hardy inequality is valid in a domain if the complement of the domain is uniformly p-fat. The same is true for the so-called pointwise Hardy inequalities as well, but for these the uniform fatness of the complement is also necessary, and so there is an equivalence between the two concepts. I will discuss this result and related Hausdorff content density conditions, and also some generalizations for weighted Hardy inequalities.

We discuss recent work with Jason Metcalfe and Daniel Tataru on local well posedness results for quasilinear Schrödinger equations. We will discuss both a natural functional framework, as well as the local smoothing, energy estimates and multilinear estimates required.

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.