We meet Thursdays at 2:45–3:45 pm in Whitney Hall 100E. This semester's organizers are James Hyde and Lorenzo Ruffoni. The seminar has an announcement mailing list open to all.
Topics include: geometric group theory, differential geometry and topology, low-dimensional topology, algebraic topology, and homotopy theory.
Abstract: Scissors congruence is the study of polytopes, up to the relation of cutting into finitely many pieces and rearranging the pieces. In the 2010s, Zakharevich defined a “higher” version of scissors congruence, where we don't just ask whether two polytopes are scissors congruent, but also how many scissors congruences there are from one polytope to another.
Zakharevich's definition is a form of algebraic K-theory, which is famously difficult to compute, but I describe some recent work that makes these calculations possible, at least for low-dimensional geometries. This allows us to compute the homology of the group of cut-and-paste operations in new cases, including the group of interval exchange transformations, and a new proof of Szymik and Wahl's theorem that Thompson's group V is acyclic.
Much of this talk is based on joint work with Anna-Marie Bohmann, Teena Gerhardt, Mona Merling, and Inna Zakharevich, and also with Alexander Kupers, Ezekiel Lemann, Jeremy Miller, and Robin Sroka.
Abstract: In the 1980s, Mahowald and Kane used integral Brown-Gitler spectra to construct splittings of the cooperations algebras for ko, connective real k-theory, and ku, connective complex k-theory. These splittings helped make it feasible to do computations using the ko- and ku-based Adams spectral sequences.
These spectral sequences have proven to be powerful tools for better understanding the structure of the stable homotopy groups of the sphere, with a variety of interesting applications. For example, Mahowald used them to verify the Telescope Conjecture at height one, and Gonzalez later used them to classify stunted lens spaces.
In this talk, I will discuss progress towards developing analogues of these tools in the C_2 equivariant setting. In particular, Guchuan Li, Sarah Petersen, and I have constructed models for C_2-equivariant analogues of the integral Brown-Gitler spectra, and used them to construct an analogue of the ku-splitting.
Abstract: Given a discrete group of isometries of hyperbolic space, we can look at its limit set, i.e., the set of accumulation points of its orbits on the sphere at infinity. This is a compact metric space embedded in the sphere at infinity, and it often displays interesting geometric and topological features that can reveal algebraic information about the group itself.
In this talk, first we will discuss the general theory and present classical examples of limit sets, including some simple fractals (Cantor set, Sierpinski carpet). Then, we will present the construction of groups whose limit set is a Pontryagin sphere. These groups are obtained as reflection groups, and the construction is based on the existence of certain right-angled hyperbolic polyhedra. This is joint work with S. Douba, G.-S. Lee, and L. Marquis.
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