Collin Bleak

My research primarily uses close analysis of the dynamics of a group or semigroup action on a space to discern properties of the group or semigroup. In particular, I am primarily interested in groups of automorphisms of a space and in semigroups of endomorphisms of a space. Supporting this, I have a secondary focus on topological and symbolic dynamics, which itself has lead me to an interest in automata and formal language theory. Most of my past and current work centers around studies of large groups of homeomorphisms. Typical groups of interest are PL(I), Homeo(S), and Homeo(C), the groups of piecewise-linear homeomorphisms of the n-cube I = [0, 1], the group of homeomorphisms of the unit circle S, and the group of homeomorphisms of the Cantor Set C. A consequence of this is that many of my results have direct bearing on the theory of the family of R. Thompson groups F ≤ T ≤ V and their relatives. In the discussion which follows, I will primarily describe the impact of my work on the family of “Thompson-like” groups. Along the way I will also mention other areas impacted by my research.