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.