We shall review recent progress obtained in the understanding of localization
properties of random Schrodinger operators. The hard issue of the Anderson
transition is stated in terms of the spreading and of the non spreading of a
wave-packet initially located at the origin. It particular it is shown that
slow transport cannot happen for ergodic random operators. As an application,
we study quantum Hall systems, that is the Hamiltonian of an electron confined
to a two dimensional plane and subjected to a constant transverse magnetic
field. We prove delocalization around each Landau level, and localization
outside a small neighborhood of these levels.

This talk will describe the simulation, design and optimization of a qubit
for use in quantum communication or quantum computation. The qubit is
realized as the spin of a single trapped electron in a semi-conductor
quantum dot. The quantum dot and a quantum wire are formed by the
combination of quantum wells and gates. The design goal for this system is a
"double pinchoff", in which there is a single trapped electron in the dot
and a single (or small number of) conduction states in the wire. Because of
considerable experimental uncertainty in the system parameters, the optimal
design should be "robust", in the sense that it is far away from
unsuccessful designs. We use a Poisson-Schrodinger model for the
electrostatic potential and electron wave function and a semi-analytic
solution of this model. Through a Monte Carlo search, aided by an analysis
of singular points on the design boundary, we find successful designs and
optimize them to achieve maximal robustness.

In this talk, I will give a survey on
some of recent advances in orbifold theory and focus
on the application. It includes the computation for
cohomology of Hilbert scheme of points of algebraic surface,
symplectic resolution, twisted K-theory and many other stuff.

I will discuss the proofs of some conjectural formulas
about Hodge integrals on moduli spaces of curves.
The generating series for all genera and all marked
points of such integrals are expressed in terms of
finite closed formulas from Chern-Simons knot invariants.
Such conjectures were made by string theorists based
on large N duality in string theory. I will explain
our proofs from localization techniques. Their relation
to toric Calabi-Yau manifolds and equivariant index
theory in gauge theory will also be discussed.
These are joint works with C.-C. Liu, J. Zhou and J. Li.

The Diederich-Fornaess worm domain has proved to be of fundamental
importance in the understanding of the geometry of pseudoconvex domains in
multidimensional complex space. More recently, the worm has proved to be
an important example for the study of the inhomogeneous Cauchy-Riemann
equations in higher dimensions.

In forthcoming work, Krantz and Marco Peloso have done a complete
analysis of the Bergman kernel on a version of the worm domain.
We produce an asymptotic expansion for the kernel and calculate
its mapping properties. We can recover versions of the results
of Kiselman, Barrett, Christ, and Ligocka on the worm.