This a joint work with Wanke Yin.
Let $M\subset \mathbb{C}^{n+1}$ ($n\ge 2$) be a real
analytic submanifold defined by an equation of the form:
$w=|z|^2+O(|z|^3)$, where we use $(z,w)\in {\CC}^{n}\times \CC$
for the coordinates of ${\CC}^{n+1}$. We first derive a pseudo-normal form
for $M$ near $0$. We then use it to prove that $(M,0)$ is holomorphically
equivalent to the quadric $(M_\infty: w=|z|^2,\ 0)$ if and only if it can
be formally transformed to $(M_\infty,0)$, using the rapid convergence
method.

Abstract: I introduce and discuss the notions of projective stability and
projective rigidity in Banach spaces, focusing on the space of integrable
functions and its noncommutative and nonassociative analogs.

Bedford and Taylor showed in 1982 that the complex
Monge-Ampere operator can be well defined (as a regular measure) for locally bounded plurisubharmonic (psh) functions,
and is continuous (in the weak topology) for decreasing
sequences. On the other hand, it is known that this operator
cannot be well defined for all psh functions. We will give a
precise characterization of its domain of definition. It turns
out that in dimension 2 it consists precisely of those psh
functions that belong to the Sobolev space W^{1,2}_{loc}.
We will also discuss a related question on compact K\"ahler
manifolds.

In this talk I will present some results concerning the
effect of large dispersion mechanism (given in the form of
$Lu_{xxx}$ or $iLu_{xx}$, where $L$ is a very large parameter) on
the long-time dynamics of dissipative evolution equations, such as
the one-dimensional complex Ginzburg-Landau and the
Kuramoto-Sivashinsky equations.

I will first give a brief introduction of the connection between
a Hamilton-Jacobi equation and the Aubrey-Mather theory. This is the so
called weak KAM theory. An extremely interesting project in weak KAM
theory is to understand what kind of dynamical information is encoded in
the
effective Hamiltonian. I will present a result about the connection
between linear pieces on level curves of the effective Hamiltonian and the
structure of correspondent Aubry sets.

We consider the homogenization for a very general class of
nonlinear, nonlocal "elliptic" equations. Motivated by the techniques
of the homogenization of fully nonlinear uniformly elliptic second order
equations by Caffarelli- Souganidis- Wang, we show how a nonlocal
version of an obstacle problem can be used to identify the effective
equation in the nonlocal setting.

"Distinguished varieties" are a special class of algebraic
curves in C^2 that exit the bidisk through the distinguished boundary
(aka the torus). We shall discuss connections with the polynomials
that define these curves and polynomials with no zeros on the bidisk,
and use a powerful "sums of squares" formula (actually a two variable
version of the Christoffel-Darboux formula for orthogonal polynomials)
to a prove a determinantal representation of distinguished varieties.
As an application of our approach, we will prove a certain bounded
analytic "extension" theorem.

The complex analogue of Diecke's Theorem and Brickell's Theorem in
real Finsler geometry. Complex Finsler structures naturally satisfy the complex
homogeneous Monge-Ampere equation and the analogue of Diecke's Theorem and
Brickell's Theorem can be put in the frame work of the classification of
complex manifolds admitting an exhaustion function satisfying the complex
homogeneous Monge-Ampere equation.

Growth estimates for orthogonal polynomials with
respect to area measure (Bergman polynomials) over the union of
finitely many Jordan regions with piecewise smooth boundary are
obtained by a careful investigation of the Green function of the
complement, and of Schwarz reflection in analytic arcs of the
boundary. As applications one derives a detailed picture of the
limiting zero distribution of Bergman's orthogonal polynomials,
and also a robust reconstruction algorithm of the
original open set, starting from incomplete data (such as obtained
by geometric tomography).