We show that the properly scaled spectral measures
of symmetric Hankel and Toeplitz matrices of size N by N generated by
i.i.d. random variables of zero mean and unit variance converge weakly
in N to universal, non-random, symmetric
distributions of unbounded support, whose moments are
given by the sum of volumes of solids related to Eulerian numbers.
The universal limiting spectral distribution for
large symmetric Markov matrices
generated by off-diagonal i.i.d. random variables
of zero mean and unit variance, is more explicit, having
a bounded smooth density given by the free convolution of the
semi-circle and normal densities.

Time permitting, I will also explain the formula for the
large deviations rate function
for the number of open path of length k in random graphs
on N>>1 vertices with
each edge chosen independently with probability 0

This talk focuses on asymptotic properties
of geometric branching processes on hyperbolic spaces
and manifolds. (In certain aspects, processes on hyperbolic spaces are
simpler than on Euclidean spaces.)
The first paper in this direction was
by Lalley and Sellke (1997) and dealt with a homogenous branching diffusion on a hyperbolic (Lobachevsky) plane).
Afterwards, Karpelevich, Pechersky and Suhov (1998) extended
it to general homogeneous branching processes on
hyperbolic spaces of any dimension. Later on, kelbert and Suhov
(2006, 2007) proceeded to include non-homogeneous branching
processes. One of the main questions here is to calculate
the Hausdorff dimension of the limiting set on the absolute.
I will not assume any preliminary knowledge of hyperbolic
geometry.

Let Fq be the finite field and : XY an Fq cover of normal varieties. We call exceptional if it maps 1-1 on Fqt points for an infinity of t. We say over Q is exceptional if it is exceptional mod infinitely many p. When X=Y, and is over Q, we have a map: exceptional p period of mod p. RSA cryptography uses x xk (k odd) and Euler's Theorem gives us its periods.

We give a paragraph of history: Schur (1921) posed a list of all Q exceptional polynomials. This inspired Davenport and Lewis (1961) to propose that a geometric property C D-L C would imply a polynomial is exceptional. Both were right (1969). Serre's O(pen) I(mage) T(heorem) produces most remaining exceptional Q rational functions (1977).

We use the D-L generalization to show exceptional covers (of Y over Fq) form a category with fiber products: the (Y,Fq) exceptional tower. Using that we can generate subtowers that connect the tower to two famous results.

I. Denef-Loeser-Nicaise motives: They attach a "motivic Poincare series" to any problem over Q. A generalization of exceptional covers produces (we say Weil) relations among Poincare series over (Y,Fq). The easiest converse question is this: If the zeta functions of X and P1 have a special Weil relation, is X an exceptional cover?

II. Serre's O(pen) I(mage) T(heorem): Rational functions from the OIT generate two (P1,Fq) exceptional subtower. The C(omplex) M(ultiplication) part of the OIT produces exceptional covers. We see their periods from the CM analog of Euler's Theorem. Periods of the subtower from the G(eneral) L(inear) part of the OIT give our most serious challenge.

In a proof-of-retrievability system, a data storage center must prove to a verifier that he is actually storing all of a client's data. The central challenge is to build systems that are both efficient and *provably* secure -- that is, it should be possible to extract the client's data from any prover that passes a verification check. All previous provably secure solutions require that a prover send O(l) authenticator values (i.e., MACs or signatures) to verify a file, for a total of O(l^2) bits of communication, where l is the security parameter. The extra cost over the ideal O(l) communication can be prohibitive in systems where a verifier needs to check many files.

We create the first compact and provably secure proof of retrievability systems. Our solutions allow for compact proofs with just one authenticator value -- in practice this can lead to proofs with as little as 40~bytes of communication. We present two solutions with similar structure. The first one is privately verifiable and builds elegantly on pseudorandom functions (PRFs); the second allows for publicly verifiable proofs and is built from the signature scheme of Boneh, Lynn, and Shacham in bilinear groups. Both solutions rely on homomorphic properties to aggregate a proof into one small authenticator value.

(Joint work with Brent Waters)

The classical Friedrichs identity states that for $u\in C_0^{\infty}(\mathbb{R}^n)$ we have
$$\int_{\mathbb{R}^n} |D^2 u|^2\, dx = \int_{\mathbb{R}^n} |\Delta u|^2\, dx.$$
From this inequality we immediately get $W^{2,2}$-estimates for
solutions of $\Delta u =f$ and also for solutions of measurable perturbations
of the form $\sum_{ij}a_{ij}(x) u_{ij}(x)=f(x)$, when the matrix
$A=(a_{ij})$ is closed to the identity in sense made precise
by Cordes.
In this talk we first explore extensions of the Friedrichs identity in
the form of sharp inequalities
$$\int_{X} |\mathfrak{X}^2 u|^2\, dx \le C_1 \int_{X} |\Delta_{\mathfrak{X}} u|^2\, dx +C_2\int_{X} |\mathfrak{X}u|^2 \, dx $$
where $X$ is a Riemannian manifold, the Heisenberg group, and certain types of CR manifolds.
\par
We then show how to use these estimates to study quasilinear subelliptic equations.\par

This is joint work with Andr\`as Domokos (PAMS 133, 2005) and Sagun
Chanillo (2007 preprint.)