Logic Seminar

The Lelek fan L is a compact and connected space with many
symmetries, which can be constructed from a projective Fraisse limit,
and hence it has a very rich homeomorphism group H(L).

In the talk, I will first show a number of properties of H(L) -- it
is totally disconnected, generated by every neighbourhood of the
identity, has a dense conjugacy class, and is simple. We then focus on
the dynamics of H(L). Using the Graham-Rothschild theorem,
the Kechris-Pestov-Todorcevic correspondence, as well as some new ideas,
we describe the universal minimal flow of H(L).  If time permits, we show
a generalization of the finite Gowers' Ramsey theorem to multiple tetris-like
operations and apply it to conclude that a group of homeomorphisms that
preserves a "typical" linear order of branches of L is extremely amenable.

This is joint work with Dana Bartosova.