Purdue Topology Seminar
In Spring 2025, the Purdue Topology Seminar will be held on Wednesdays 11:30am - 12:30pm EST at BRNG B206 (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.
Spring 2025
January 15 (in person)
Title: An equivariant pair-of-paints product from higher Hochschild cochains
Abstract: Hochschild homology and cohomology are both generally defined in the settings of A-infinity algebras/categories. In general, Hochschild homology does not inherently possess a ring structure, unlike its cohomological counterpart. Rather, it is a module over Hochschild cohomology. However, when a (homologically) smooth A-infinity algebra/category, introduced by Kontsevich-Soibelman, is equipped with a smooth Calabi-Yau structure, a product structure emerges on Hochschild homology, introducing a richer algebraic framework. In the presence of group actions (for instance, Z/2 in our case), a natural question arises: ``Can this (non-equivariant) product structure, called pair-of-pants product, often be extended or adapted to its equivariant counterpart?’'.
In this talk, I will show that there is a canonical equivariant refinement, called algebraic equivariant pair-of-pants product, of the (non-equivariant) product structure provided that a canonical lift, of a Calabi-Yau map, induced by a (strong) smooth Calabi-Yau structure exists. I will also give the motivations behind the question via a conjectural diagram related to open-closed maps, fixed point Floer cohomology and Hochschild homology of Fukaya categories and Smith-type inequalities (modulo technical details) as long as time permits. This is joint work with Sheel Ganatra.
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)
Title: McDuff's secondary class and the Euler class of foliated sphere bundles
Abstract: Tsuboi proved that the Calabi invariant transgresses to the Euler class of foliated circle bundles. The Calabi invariant is a first cohomology class of the group of area-preserving diffeomorphisms of a closed disk that are identity on the boundary.
In this talk, I will introduce a group cocycle representative of McDuff's secondary class: a (2n−1)-dimensional cocycle of the group of volume-preserving diffeomorphisms of a 2n-dimensional closed ball that are identity on the boundary. I will also discuss a higher-dimensional analog of Tsuboi's theorem, which states that this cohomology class transgresses to the Euler class of foliated sphere bundles.
February 6 (this is a Tuesday, time and location TBD, in person)
February 19 (in person)
Calvin McPhail-Snyder (Duke University)
March 12 (online)
Amit Kumar (Louisiana State University, IMSc)