Professor Steve Jackson invites you to attend the PhD dissertation defense of Edward Krohne today, March 22nd at 3:00 pm in GAB 461. Cake and coffee will be served in GAB 472 following this event.

"Continuous combinatorics on F(2(Z2))"

Abstract:

We present a novel theorem of Borel Combinatorics that sheds light on the types of continuous functions that can be defined on the graph of F(2(Z2)). The topological space F(2(Z2)) embeds into the Cantor space 2ω and has a natural free continuous Z2 action. Considering the graph induced by this action, we obtain a disjoint union of uncountably many Cayley graphs of Z2. It is folklore that no continuous (indeed, Borel) function provides a chromatic two-coloring of F(2(Z2)), despite the fact that any finite part of the graph on F(2(Z2)) is bipartite. Our main result offers a much more complete analysis of continuous functions on this space. We construct a countable collection of finite graphs Γ(n,p,q) (with n≤p⊥q≤ω) each consisting of twelve "tiles," such that for any property P (such as "chromatic two-coloring") that is locally recognizable in the proper sense, a continuous function with property P exists on F(2(Z2)) iff a function with a corresponding property P0 exists on some Γ(n,p,q). We present the theorem, and give several applications.