Purdue Topology Seminar
On Fall 2024, the Purdue Topology Seminar will be held on Wednesdays 10:30am - 11:30am EST at BHEE 236 (if we meet in person) unless otherwise noted. Some of the talks will be online through Zoom. If you want to be added to our email list please contact Manuel Rivera (manuelr at purdue.edu).
We have begun to record some of our online seminars. We publish them on our YouTube-Channel.
Fall 2024
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
November 13 (in person)
Ralph Kaufmann (Purdue University)
Title: A bimodule approach to Categories and Hopf algebras I
Abstract: We will discuss new results about Feynman categories and their appearance in Hopf algebras. One of these is joint with Micheal Monaco and Yang Mo about three different bar/cobar resolutions of ops generalizing the work of Muriel Livernet for operads. The second is the Koszul duality in the framework of Feynman categories, especially the Schwarz Modular operads. This is joint with Ben Ward. The third are new connections of the Hopf algebras from Feynman categories to field theory and Waldhausen K-theory. The background with which we will start is a way to think about categories as monoidal bi-module monoids in a monoidal functor category.
November 20 (in person)
Michael Monaco (Purdue University)
Title: A bimodule approach to Categories and Hopf algebras II
Abstract: We will give technical details about the constructions introduced in the first part (November 13th).
December 4 (online)
Lea Kenigsberg (UC Davis)
Title: The non-homotopy invariance of the string topology coproduct via fixed point traces
Abstract: Let M be a smooth manifold, for example a higher dimensional sphere. The free loop space is the space of all maps from the circle to M. The loop space admits a very rich structure which reveals geometric and topological information about manifolds. In this talk I will describe the string topology coproduct and the kind of information it encodes. In particular, I will describe its failure to be a homotopy invariant via connections to fixed point theory. This is based on joint work with Noah Porcelli.
December 11 (in person)
Talk 1: 9:30 - 10:30
Title: Galois Symmetry on Manifold Structures of Varieties
Abstract: It is known that there are complex varieties which are algebraically isomorphic but not homeomorphic. In particular, the homeomorphism types might change under the Galois automorphisms of the ground field. However, the underlying finite homotopy information is Galois invariant. In this talk, I will introduce some results in studying how the underlying manifold structures change under Galois automorphisms. There are three aspects of this: 1) how to formalize manifolds by topological invariants; 2) what is the geometric aspect of finite homotopy information; 3) how to formulate compatible Galois symmetry on formal manifolds.
Talk 2 10:30 - 11:30 am
Title: Galois Symmetry on Manifold Structures of Varieties
Abstract: It is known that there are complex varieties which are algebraically isomorphic but not homeomorphic. In particular, the homeomorphism types might change under the Galois automorphisms of the ground field. However, the underlying finite homotopy information is Galois invariant. In this talk, I will introduce some results in studying how the underlying manifold structures change under Galois automorphisms. There are three aspects of this: 1) how to formalize manifolds by topological invariants; 2) what is the geometric aspect of finite homotopy information; 3) how to formulate compatible Galois symmetry on formal manifolds.
Spring 2025
January 15 (in person)
January 22 (online)
Maximilian Stegemeyer (Freiburg)
Title: Intersection multiplicity in loop spaces and the string topology coproduct
Abstract: String topology is the study of algebraic structures on the free loop space of a closed oriented manifold. The Goresky-Hingston coproduct is geometrically defined by cutting loops apart at points of self-intersection. A result by Hingston and Wahl makes this intuitive description precise and sates that if a homology class can be represented by loops without basepoint self-intersections then the coproduct of this class vanishes. In this talk I will talk about some results where the concept of intersection multiplicity is used to conclude the triviality of the string topology coproduct. We will also encounter the relation between the homology of the free loop space and the closed geodesics in the underlying manifold. The triviality results for the coproduct open up a number of questions.
January 29 (online)
Shuhei Maruyama (Kanazawa University)