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 $F(2^(Z^2 ) )$"

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 $F(2^(Z^2 ) )$. The topological space $F(2^(Z^2 ) )$ embeds into the Cantor space $2^ω$ and has a natural free continuous $Z^2$ action. Considering the graph induced by this action, we obtain a disjoint union of uncountably many Cayley graphs of $Z^2$. It is folklore that no continuous (indeed, Borel) function provides a chromatic two-coloring of $F(2^(Z^2 ) )$, despite the fact that any finite part of the graph on $F(2^(Z^2 ) )$ 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 $F(2^(Z^2 ) )$ iff a function with a corresponding property $P_0$ exists on some $Γ_(n,p,q)$. We present the theorem, and give several applications.