In this talk I shall provide a survey of my recent works and their environs on differential geometry of Lagrangian submanifolds in specific K\"ahler manifolds, such as complex projective spaces, complex space forms, Hermitian symmetric spaces and so on. I shall emphasis on the relationship between certain minimal Lagrangian submanifold in complex hyperquadrics and isoparametric hypersurfaces in spheres. This talk is mainly based on my joint work with Associate Professor Hui Ma (Tsinghua University, Beijing).

We study some problems relating to polynomials evaluated either at random Gaussian or random Bernoulli inputs.  We present a structure theorem for degree-d polynomials with Gaussian inputs. In particular, if p is a given degree-d polynomial, then p can be written in terms of some bounded number of other polynomials q_1,...,q_m so that the joint probability density function of q_1(G),...,q_m(G) is close to being bounded.  This says essentially that any abnormalities in the distribution of p(G) can be explained by the way in which p decomposes into the q_i.  We then present some applications of this result.

We introduce a new class of spaces of mixed Hardy-Bergman type on 

a half-plane.  We study some functional properties of these spaces, and their

zero-sets.  We apply these results to the Muntz-Szasz problem for the Bergman

space.  (This is joint work, in progress, with M. Salvatori.

 I will outline a construction of an exotic solution of the nonlinear
Schrödinger equation that exhibits a big frequency cascade. Recent advances
related to this construction and some open questions will be surveyed.

Eigenfunctions of the Laplacian on a bounded domain represent the modes of vibration of a vibrating drum. The behavior of these eigenfunctions is closely related to the behavior of the underlying dynamical system of the billiard table. In this talk I first give a brief exposition on this relation and then I talk about the boundary traces of eigenfunctions and a recent joint work with Han, Hassell and Zelditch.