Schrodinger flow is the Hamiltonian flow for energy functional on the space of maps from a Riemannian manifold into a Kahler manifold. I'll talk about some background on this flow, then focus on the special case of maps from a Euclidean space into the complex Grassmannian Gr(k,C^n). Terng and Uhlenbeck proved that Schrodinger flow of maps from R^1 into complex Grassmannian is gauge equivalent to the matrix nonlinear Schrodinger equation. Using this gauge equivalence and the result of Beals and Coifman, they obtained the global existence of Schrodinger flow with rapidly decay initial data. Applying the method of Terng and Uhlenbeck, we will see that Schrodinger flow of radial maps from R^m into the complex Grassmannian is gauge equivalent to a generalized matrix nonlinear Schrodinger equation. When the target is the 2-sphere, the gauge equivalence was studied by Lakshmanan and his colleagues by different method. They also observed that if the domain is R^2, then the corresponding matrix nonlinear Schrodinger equation is an integrable system.

We present some new results concerning the Dirichlet problem for the prescribed mean curvature equation over a bounded domain in R^n. In the case when the mean curvature is zero this can be posed variationally as the problem of finding a least area representative among functions of bounded variation with prescribed boundary values. We show that there is always a minimizer which is represented by a compact C^{1,alpha} manifold with boundary, with boundary given by the prescribed Dirichlet data, provided this data is C^{1,alpha} and it is of class C^{1,1} if the prescribed data is C^3.

Consider a family of smooth compact connected $n$ dimensional Riemannian manifolds. What can one say about the spectral geometry of a limit of these?

This question has interested many spectral geometers; my talk focuses on conical metric degeneration in which the family converges "asymptotically conically'' to an open manifold with conical singularity. I will present spectral convergence results and discuss techniques including microlocal analysis on manifolds with corners and geometric blowup constructions. I will also summarize spectral convergence results for other geometric contexts and discuss applications and open questions.