This talk will discuss random walks on percolation clusters.

The first case is supercritical ($p>p_c$) bond percolation in
$Z^d$. Here one can obtain Aronsen type bounds on the transition
probabilities, using analytic methods based on ideas of Nash.

For the critical case ($p=p_c$) one needs to study the incipient
infinite cluster (IIC). The easiest situation is the IIC on trees -
where the methods described above lead to an alternative approach to
results of Kesten (1986). (This case is joint work with T. Kumagai).

We discuss how to use irreducible polynomials with big Galois groups
in order to construct abelian varieties without nontrivial
endomorphisms.

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.

We consider Jacobi matrices built on equilibrium measures of hyperbolic polynomials. We show their property, which, on one side, is related to almost periodicity of such matrices, and, on the other side, is a sort of noncommutative
Perron-Frobenius-Ruelle theorem. While proving these key property one is naturally brought to consider a two-weight Hilbert transform. Its boundedness can be proved in our situation, while the general two-weight Hilbert transform
boundedness criterion is not yet available.
We will mention other problems in spectral theory of Jacobi matrices, where this paradigm of nonhomogeneous harmonic analysis---two weight Hilbert transform---appears in the natural way.

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.