We consider the spectrum of the Fibonacci Hamiltonian for small
values of the coupling constant. It is known that this set is a Cantor set
of zero Lebesgue measure. We show that as the value of the coupling constant
approaches zero, the thickness of this Cantor set tends to infinity, and,
consequently, its Hausdorff dimension tends to one. Moreover, the length of
every gap tends to zero linearly. Finally, for sufficiently small coupling,
the sum of the spectrum with itself is an interval. The last result provides
a rigorous explanation of a phenomenon for the Fibonacci square lattice
discovered numerically by Even-Dar Mandel and Lifshitz. The proof is based
on a detailed study of the dynamics of the so called trace map. This is a
joint work with David Damanik.

The local Langlands correspondence for GSp(4)
gives a classification of irreducible complex representations of GSp(4,k),
where k is a p-adic field in terms of 4-dimensional symplectic
Galois representations (plus some additional data). I will describe the
precise statement and give an idea of its proof. I will also mention
some further questions in this direction. This is joint work with
Shuichiro Takeda.

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. We also use it to give a necessary and sufficient condition
when $(M,0)$ can be formally flattened. Our main theorem generalizes a
classical result of Moser for the case of $n=1$.

Starting from a nondecreasing function $K:[0,\infty)\to [0,\infty)$,
we consider a M\"obius-invariant Banach space $Q_K$ of functions
analytic in the unit disk. For $0

This talk is for a more sophisticated audience, going over Perelman's noncollapsing estimate for the adjoint heat equation and my nonconic estimate.