In this talk, we study Cauchy-Riemann equation on some smooth convex domains of innite type in
C2. In detail, we show that supnorm estimates hold for those infinite exponential type
domains provided the exponent is less than 1. This is a joint work with John Erik Fornaess
and Lina Lee.

I will present elementary (classical) proof(s) that every derivation of a
finite dimensional semisimple algebra (associative, Lie, or Jordan) is
inner, and state what is known in infinite dimensions for operator
algebras. Then I will do the same for the corresponding triple systems.
The purpose is to set the stage for the study of continuous triple cohomology.

We discuss some resent results for two type of spectral inequalities in connection with the Navier-Stokes equations. Namely, the Berezin-Li-Yau inequalities for the eigenvalues of elliptic equations and systems ( including the Stokes system) with constant coefficients and the Lieb-Thirring inequalities for the negative spectrum of the Schrodinger operators. In the one-dimensional periodic case we obtain a simultaneous bound for the negative trace and the number of negative eigenvalues.

We consider the global embedding problem for compact, three dimensional
CR manifolds. Sufficient conditions for embeddability are obtained from assumptions on the CR Yamabe
invariant and the non-negativity of a certain conformally invariant fourth order operator called the CR Paneitz
operator. The conditions are shown to be necessary for small deformations of the standard CR structure on the three sphere.

Composition operators in one variable have been studied very extensively.
But in several variables case the progress is very slow and even the boundedness of
the composition operator on the unit ball is not characterized yet, except the Carleson measure type characterization.
We survey the progress on the composition operators on the unit ball and discuss
some open problems.

We define the Szego metric using the Szego kernel and Fefferman surface measure. This metric is invariant under biholomorphic mappings. We compare this metric with Caratheodory and Bergman metrics and also show that one can determine whether a strongly pseudoconvex is biholomorphic to a ball by studying the ratio of the Szego and Bergman metric. This is a joint work with David Barrett.

In this talk, I will review the regularity problem for
Korteweg-de Vries (KdV) equation on the line, and give a brief summary of
the sharp well-posedness and ill-posedness results. Then I will discuss a
possible way to get a-priori bounds and weak solution below the critical
threshold H^{-3/4}.

We investigate the transversality of holomorphic mappings between CR submanifolds of complex spaces. In equidimension case, we show that a holomorphic mapping sending one generic submanifold into another of the same dimension is CR transversal to the target submanifold, provided that the source manifold is of finite type and the map is of generic full rank. In different dimensions, we will show that under certain restrictions on the dimensions and the rank of Levi forms, the mappings whose set of degenerate rank is of codimension at least 2 is transversal to the target.

In this lecture, we will talk about a recent joint
work ofGordon Heier and myself about curvature characterizations
ofuniruledness and rational connectivity of projective manifolds. A
result on projective manifolds with zero total scalar curvature will
also be discusse.

The classical Cauchy integral is a fundamental object of complex analysis whose analytic properties are intimately related to the geometric properties of its supporting curve.

In this talk I will begin by reviewing the most relevant features of the classical Cauchy integral. I will then move on to the (surprisingly more involved) construction of the Cauchy integral for a hypersurface in
$\mathbb C^n$.

I will conclude by presenting new results joint with E. M.