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.

It is well known that the square root of any integer can be written as a linear combination of roots of unity. A generalization of this fact is the "Kronecker-Weber Theorem", which states that in fact any element which generates an abelian Galois extension of the field of rational numbers Q can also be written as such a linear combination. The roots of unity may by viewed as the special values of the analytic function e(x) = exp(2*pi*i*x) where x is taken to be a rational number. Broadly speaking, Hilbert's 12th problem is to find an analogous result when Q is replaced by a general algebraic number field F, and in particular to find the analytic functions which play the role of e(x) in this general setting.

Hilbert's 12th problem has been solved in the case where F is an imaginary quadratic field, with the role of e(x) being played by certain modular forms. All other cases are, generally speaking, unresolved. In this talk I will discuss the case where F is a real quadratic field, and more generally, a totally real field. I will describe relevant conjectures of Stark and Gross, as well as current work using a p-adic approach and methods of Shintani. A proof of these conjectures would arguably provide a positive resolution of Hilbert's 12th problem in these cases.

In this talk, we will discuss the existence and the
compactness results of a class of fully nonlinear PDEs as well as
their applications.

I will be discussing ideas to enrich the undergraduate program in mathematics at UCI. First, I will present ideas about standardizing the testing, and the content, of the large service courses such as Math 1A-B, 2A-B, as well as perhaps Math 7. This would involve the use of the state-of-the-art technology and the creation and administration of common exams. Second, I will discuss my thoughts about deepening the involvement of the mathematics department in community outreach programs. This would involve coordinating with programs such as CAMP and MESA. Third, I will present my ideas about enhancing our honors program and upper division courses, as well as enriching the undergraduate math club. And lastly, I will discuss ways of enticing students into doing more research. This would involve finding interested students, matching them with appropriate faculty, and working with programs such as UROP.

The Langlands conjecture originated as a highly non-trivial generalization of the reciprocity laws in number theory. In my talk, I explain how after certain `geometrization', it becomes a statement about sets (`moduli spaces') of vector bundles on a Riemann surface. The result is a kind of Fourier transform relating sets of vector bundles and local systems on a Riemann surface.

This `geometric Langlands transform' can be used to motivate theorems and conjectures in such diverse areas of mathematics (and physics) as theory of Painleve equations, representation theory of loop groups, autoduality of Jacobians, and mirror symmetry. Some of the relations will be explained in the talk.