Keynote Speaker
Eric Swenson
Brigham Young University
From continua to trees
Abstract: Let Z be a continuum, a compact connected metric space. Z is called m-thick if Z can be separated by removing m points, but cannot be separated by removing less than m points. A subset A ⊆ Z of an m-thick continuum Z is called a min-cut if Z-A is not connected and |A|=m. We will see that for any m-thick continuum there is an ℝ tree T which encodes the separation "structure" of the min-cuts of Z. What this is used for: If you have a group G acting on Z by homeomorphisms, you obtain an action of G on T by homeomorphisms. Under certain conditions on the action of G on Z you obtain a non-nesting action of G on T. This allows us to use the Rips machine to understand the structure of G. | Keynote Speaker
Tim Riley
Cornell University
The geometry of the Conjugacy Problem
Abstract: In 1910 Max Dehn posed three problems he saw as fundamental for understanding groups: the Word Problem, the Conjugacy Problem, and the Isomorphism Problem. The Word Problem is the challenge of finding an algorithm which, given a word w on a given group's generating set, declares whether or not that word represents the identity. This relates to isoperimetry: the `soap-film geometry' of minimal area disc-fillings of loops in spaces. The Conjugacy Problem is the challenge of recognizing when two words represent conjugate elements in a group. It too has a geometric interpretation, but concerning annuli rather than discs. I will survey some results and questions that arise. | Keynote Speaker
Zoran Šunić
Hofstra University
Ordering free groups, trees, and free products, not necessarily in that order
Abstract: We introduce left orders on the free group of any finite rank that are particularly easy to state and work with. The original approach uses actions of the free group on the real line, which is itself an ordered object, and then letting the group just "borrow" the order from the line. We then recover the same orders, and a lot more, through a more organized and sophisticated approach. Namely, we order free products by first ordering their associated Bass-Serre trees and then letting the groups "borrow" the order from the trees. |