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.

The geometric theory of Banach spaces underwent a tremendous development in the decade 1990-2000 with the solution of several outstanding conjectures by Gowers, Maurey, Odell and Schlumprecht.

Their discoveries both hinted at a previously unknown richness of the class of separable Banach spaces and also laid the beginnings of a classification program for separable Banach spaces due to Gowers.

However, since the initial steps done by Gowers, little progress was made on the classification program. We shall discuss some recent advances due to V. Ferenczi and myself on this by means of Ramsey theory and dichotomy theorems for the structure of Banach spaces. This simultaneously allows us to answer some related questions of Gowers concerning the quasiorder of subspaces of a Banach space under the relation of isomorphic embeddability.

The geometric theory of Banach spaces underwent a tremendous
development in the decade 1990-2000 with the solution of several
outstanding conjectures by Gowers, Maurey, Odell and Schlumprecht.

Their discoveries both hinted at a previously unknown richness of the
class of separable Banach spaces and also laid the beginnings of a
classification program for separable Banach spaces due to Gowers.

However, since the initial steps done by Gowers, little progress was
made on the classification program. We shall discuss some recent
advances due to V. Ferenczi and myself on this by means of Ramsey theory
and dichotomy theorems for the structure of Banach spaces. This
simultaneously allows us to answer some related questions of Gowers
concerning the quasiorder of subspaces of a Banach space under the
relation of isomorphic embeddability.

We study nonlinear integro-differential equations. Typical examples are the ones that arise from stochastic control problems with discontinuous Levy processes. We can think of these as nonlinear equations of fractional order. Indeed, second order elliptic PDEs are limit cases for integro-differential equations. Our aim is to extend the theory of fully nonlinear elliptic equations to this class of equations. We are able to obtain a result analogous to the Alexandroff estimate, Harnack inequality and $C^{1,\alpha}$ regularity. As the order of the equation approaches two, in the limit our estimates become the usual regularity estimates for second order elliptic pdes. This is a joint work with Luis Caffarelli.