In 1969, Hirsch posed the following problem: given a diffeomorphism of a manifold and a hyperbolic set for the diffeomorphism, describe the topology of the hyperbolic set and the dynamics of the diffeomorphism for this set. We solve the problem when the hyperbolic set is a closed 3-manifold.

One possible approach to the study of the geometry of the Gibbs measure in the Sherrington-Kirkpatrick
type models (for example, the chaos and ultrametricity problems) is based on the analysis of the free energy
on several replicas of the system under some constraints on the distances between replicas. In general, this
approach runs into serious technical difficulties, but we were able to make some progress in the setting of the
spherical p-spin SK models where many computations become more explicit.

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.