Abstract: The Houghton groups $H_n$ are a family of groups that are straightforward to define but have a variety of bizarre and interesting properties. In this talk I will discuss my recent computation of the Bieri-Neumann-Strebel-Renz invariants $\Sigma^m(H_n)$ of the $H_n$. The computation reveals some geometry reminiscent of that expected for metabelian groups by Bieri's $\Sigma^m$-Conjecture, and has implications for the finiteness properties of certain subgroups of $H_n$. This talk will be self-contained, and I will not assume any particular familiarity with the Houghton groups or the BNSR-invariants.
Abstract: In order to discuss recent work, I will introduce Legendrian contact homology as formulated by Chekanov.
Abstract: The notion of a 2-Segal object was recently defined by Dyckerhoff and Kapranov, and independently by Gálvez-Carrillo, Kock, and Tonks under the name of decomposition space. Whereas 1-Segal sets model the structure of a category, in which composition is defined and is associative, 2-Segal sets instead encode a more general structure in which composition need not exist or be unique, but is still associative when it is defined. The 2-Segal set associated to a graph gives a nice example where maps can be composed in different ways. In particular, following a definition of Dyckerhoff and Kapranov, this 2-Segal set has an associated Hall algebra which is much smaller than most natural examples of such algebras and has a curious description as a cohomology ring.