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.

Tiling planforms dominated by diamonds (such as the diamond-shaped seeds on a sunflower head), hexagons, or ridges (such as those on saguaro cacti) are observed on many plants. We analyze PDE models for the formation of these patterns that incorporate the effects of growth and biophysical and biochemical mechanisms. The aim is to understand both the underlying symmetries and the information specific to the mechanisms. The patterns are compared to Voronoi tessellations, and we will start to draw a bigger picture of growth and symmetry in biological systems.