In this talk I will describe the recent results on Betti tables of graded modules.
The Betti table describe numerical data related to a minimal free resolutions of a module.
The basic idea goes back to Hilbert who first proved existence of finite free resolutions.
Recently Boij and Soderberg made striking conjectures about the general shapes of Betti tables.
It allows to say (up to an integer multiple) which Betti tables actually exist.
These conjectures were subsequently proved by Eisenbud and Schreyer.

I will define all the basic notions concerning resolutions and Betti tables, no knowledge
of these questions will be assumed.

Uniformly hyperbolic systems are nowadays fairly well understood,
both from the topological and the ergodic point of view. Outside the hyperbolic
domain, two main phenomena occur: homoclinic tangencies and cycles involving
saddles with different indices. Homoclinic classes and chain compoments are the
natural candidates to replace hyperbolic basic sets in non-hyperbolic theory. Several
recent papers explore their ”hyperbolic-like” properties, many of which hold only
for generic dynamical systems. In this talk, we study how a C1-robust dynamic
property (i.e. a property that holds for a system and all C1 nearby ones) on the
underlying manifold would influence the behavior of the tangent map on the tangent
bundle.

The current financial crisis is forcing thorough overhaul of not only the
practice but the theoretical framework of modern finance. We will talk
about how and why such crisis occurred and what kind of inadvertent, albeit
supporting, role modern finance played in creating it.. We will also
discuss some possible directions modern finance may go in. Our view is a
socio-historical as well mathematical financial one.