Slawomir Solecki: Projective Fraisse Theory

We will describe a dualization of the Fraisse limit construction from Model Theory. We will show how to represent various interesting compact spaces as canonical quotients of such projective Fraisse limits. We will finish with some applications.

Slawomir Solecki: Ramsey and Ultrafilters

We will start with stating precisely two concrete Ramsey results of the type we are interested in: Gowers' tetris theorem and Furstenberg and Katznelson's generalization of the Hales--Jewett theorem. We will indicate a general understanding of such results in terms of partial semigroups. At this point, we will introduce new algebraic structures appropriate for formalizing such theorems. This will involve in a crucial way ultrafilter spaces over partial semigroups. Then we will state a general Ramsey theorem on the existence of appropriately defined basic sequences for such structures. We will proceed to describing ways of producing such algebraic structures that are relevant to proving Ramsey theorems. This will involve (1) establishing a theorem on the structure of monoid actions by continuous endomorphisms on compact left-topological semigroups and (2) defining the operation of tensor product.