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.

We consider random matrices associated to random walks on the complete
graph with random weights. When the weights have finite second moment we
find Wigner-like behavior for the empirical spectral density. If the
weights have finite fourth moment we prove convergence of extremal
eigenvalues to the edge of the semi-circle law. The case of weights with
infinite second moment is also considered. In this case we prove
convergence of the spectral density on a suitable scale and the limiting
measure is characterized in terms certain Poisson weighted infinite
trees associated to the starting graph. Connections with recent work on
random matrices with i.i.d. heavy-tailed entries and several open
problems are also discussed. This is recent work in collaboration with
D. Chafai and C. Bordenave (from Univ. P.Sabatier, Toulouse - France).