Abstract: consider a finite graph, with an actor sitting at each node, and a
dollar on each edge. Negotiations will be conducted between pairs of
adjacent actors over splitting the dollar on the intervening edge.
At the end of negotiations, each actor may sign at most one contract with a
neighbour, agreeing on some possibly uneven split of the dollar.
How much money is each actor likely to receive? And which matchings of the
graph are likely to arise?
Kleinberg and Tardos analysed the limiting answer - a balanced solution -
that arises from assuming that actors iteratively revise current deals using
Nash bargaining, taking the best alternative deal currently available as a
backup.

Most of the talk will be expository, I'll explain the concepts of Nash bargaining and balanced solution. If there is time, I will discuss
the rate of
convergence to the balanced solution of this type of negotiation.

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.