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.
