September 11 (in-person)

Sam Nariman (Purdue University)

Title: Invariants of foliated sphere bundles

Abstract: Morita showed that for each power of the Euler class, there are examples of flat circle bundles for which the power of the Euler class does not vanish. Haefliger asked if the same holds for flat odd-dimensional sphere bundles. We will sketch that for a manifold M with free torus actions M-bundles are cobordant to a flat M-bundle and as a consequence, we answer Haefliger's question about the nonvanishing of the powers of the Euler class. We show that the powers of the Euler class and Pontryagin classes are all non-trivial for flat sphere bundles and we also talk about a recent result f Nils Prigge for flat S^3 bundles.

September 25 (in-person)

Ezekiel Lemann (Binghamton University)

Title: Scissors Automorphism Groups and Homological Stability

Abstract: In this talk we will outline the proof of homological stability for scissors automorphism groups and highlight a number of consequences and related results. These include plus and group completion constructions for assembler K-theory, interpreting higher K-theory in terms of automorphism groups, and homology calculations for some scissors automorphism groups. The content of the talk is joint work with Kupers, Malkiewich, Miller, and Sroka.

October 2 (in-person)

Sarah Anderson (Purdue University)

Title: Configuration spaces (advanced topics exam)

Abstract: This talk will give an introduction to configuration spaces, homological stability and the scanning map.

October 9 (in-person)

Nicolas Bridges (Purdue University)

Title: An Extension of the Kuperberg Invariant for Three-Manifolds from Involutory Hopf Algebras

Abstract:  We introduce an invariant of a pair (M,L) consisting of a closed connected oriented three-manifold and a framed oriented link embedded in M, which is a generalization of both the Kuperberg invariant and the Hennings-Kauffman-Radford (HKR) invariant. The invariant has as input the data of an involutory Hopf algebra and a collection of representations of the Drinfeld double. More precisely, we show that if L is the empty link, then the invariant is equivalent to the Kuperberg invariant for the manifold M, and if M is the 3-sphere and the representations are the left regular representations, then the invariant is equivalent to the HKR invariant. We also show that the invariant of (M, L) is equal to that of the 3-manifold obtained by surgery on L. This provides a new proof to a conjecture/theorem relating the Kuperberg invariant and the HKR invariant in the involutary case.

October 16 (online)

Lyne Moser (University of Regensburg) 

Title: New methods to construct model categories

Abstract: Model categories provide a good environment to do homotopy theory. While weak equivalences are the main players in a model category and encode how two objects should be thought of as being ``the same'', the additional data of cofibrations and fibrations typically facilitates computations of homotopy (co)limits and derived functors. However, because of their robust structure, model categories are usually hard to construct. In joint work with Guetta, Sarazola, and Verdugo, we develop new techniques for constructing model structures from given classes of cofibrations, fibrant objects, and weak equivalences between them. The requirement that one only needs to provide a class of weak equivalences between fibrant objects seems more natural in practice as the fibrant objects are often the ``well-behaved'' objects and so weak equivalences should only be expected to exhibit a good behavior between these objects. As a straightforward consequence of our result, we obtain a more general version of the usual right-induction theorem along an adjunction, where fibrations and weak equivalences are now only right-induced between fibrant objects. If time permits, I will mention some applications of these new methods.

November 6 (online)

Solomon Jekel (Northeastern University) 

Title: Torsion at the Threshold for Mapping Class Groups