I will try to give a broad review of the amasing
developments in the theory of infinite groups during the last 25 years. These include the
emergence of Monsters and flourishing of the Asymptotic Theory of Finite Groups. We will focus on important examples and formulate some open problems.

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.