Tuesdays at 10:15
CM 1 113
Program
Date  Title  Speaker 

25.09.2018
in CM 012 
The loop homology algebra of discrete torsion  Yasuhiko Asao
University of Tokyo 
09.10.2018  Topological Hochschild Homology and Fixed Point Invariants  Jonathan Campbell
Vanderbilt University 
16.10.2018  KTheory and Polytopes  Jonathan Campbell
Vanderbilt University 
23.10.2018  Toward the equivariant stable parametrized hcobordism theorem  Mona Merling
University of Pennsylvania 
30.10.2018  Tools for understanding topological coHochschild homology  Anna Marie Bohmann
Vanderbilt University 
06.11.2018  Thom spectra and CalabiYau algebras  Inbar Klang
EPFL 
13.11.2018 
Periodic orbits and topological restriction homology

Cary Malkiewich
Binghamton University 
20.11.2018  Arpon Raksit
Stanford University 

27.11.2018  Calista Bernard
Stanford University 

04.12.2018  
11.12..2018  
18.12.2018  Martina Rovelli
Johns Hopkins University 

05.03.2019  Sune Precht Reeh
Universitat Autònoma de Barcelona 
(See also the program of the topology seminar in 2011/12, 2010/11, 2009/10, 2008/09, 2007/08, 2006/07, and 2005/06.)
Abstracts
Yasuhiko Asao – The loop homology algebra of discrete torsion
Let $M$ be a closed oriented manifold with a finite group action by $G$.
We denote its Borel construction by $M_{G}$. As an extension of string
topology due to ChasSullivan, LupercioUribeXicot$\’{e}$ncatl
constructed a graded commutative associative product (loop product) on $
H_{*}(LM_{G})$, which plays a significant role in the “orbifold string
topology” . They also showed that the constructed loop product is an
orbifold invariant. In this talk, we describe the orbifold loop product
by determining its “twisting” out of the ordinary loop product in term
of the group cohomology of $G$, when the action is homotopically trivial.
Through this description, the orbifold loop homology algebra can be
seen as R. Kauffmann’s “algebra of discrete torsion”, which is a group
quotient object of Frobenius algebra. As a cororally, we see that the
orbifold loop product is a nontrivial orbifold invariant.
Jonathan Campbell – Topological Hochschild Homology and Fixed Point Invariants
Fixed point theorists have an array of invariants for determining if a selfmap is homotopic to one without fixed points – most are elaborations of the classical Lefschetz invariant. In this talk, I’ll discuss these invariants and show that they arise very naturally from topological Hochschild homology. Furthermore, the invariants are in the image on components of the cyclotomic trace from Ktheory to THH. This suggests higher Ktheory and THH as homes for more refined fixedpoint data. In order to make the talk accessible I’ll define the objects involved – a knowledge of the stable homotopy category is the only prerequisite. This is joint work with Kate Ponto.
Jonathan Campbell – KTheory and Polytopes
In this talk I’ll describe joint work in progress with Inna Zakharevich to understand the relationship between algebraic Ktheory and the scissors congruence problem. I’ll define both algebraic Ktheory and the scissors congruence problem, and then describe various ways of using the former to attack the latter. And possible ways that the latter can shed light on the former.
Mona Merling – Toward the equivariant stable parametrized hcobordism theorem
Waldhausen’s introduction of Atheory of spaces revolutionized the early study of pseudoisotopy theory. Waldhausen proved that the Atheory of a manifold splits as its suspension spectrum and a factor Wh(M) whose first delooping is the space of stable hcobordisms, and its second delooping is the space of stable pseudoisotopies. I will describe a joint project with C. Malkiewich aimed at telling the equivariant story if one starts with a manifold M with group action by a finite group G.
Anna Marie Bohmann – Tools for understanding topological coHochschild homology
Hochschild homology is a classical invariant of algebras. A “topological” version, called THH, has important connections to algebraic Ktheory, Waldhausen’s Atheory, and free loop spaces. For coalgebras, there is a dual invariant called “coHochschild homology” and Hess and Shipley have recently defined a topological version called “coTHH,” which also has connections to Ktheory, Atheory and free loops spaces. In this talk, I’ll talk about coTHH (and THH) are defined and then discuss work with Gerhardt, Hogenhaven, Shipley and Ziegenhagen in which we develop some computational tools for approaching coTHH.
Inbar Klang – Thom spectra and CalabiYau algebras
I’ll briefly introduce topological field theories, along with some examples of 2dimensional ones. I’ll discuss how 2dimensional TFTs are related to CalabiYau algebras, which are algebras that satisfy a certain dualizability condition. Then I will talk about joint work with Ralph Cohen, in which we defined and studied a spectrumlevel generalization of the CalabiYau condition, as well as its relation to field theories and to symplectic geometry.
Cary Malkiewich – Periodic orbits and topological restriction homology
I will talk about a project to import trace methods, usually reserved for algebraic Ktheory computations, into the study of periodic orbits of continuous dynamical systems (and viceversa). Our main result so far is that a certain fixedpoint invariant built using equivariant spectra can be “unwound” into a more classical invariant that detects periodic orbits. As a simple consequence, periodicpoint problems (i.e. finding a homotopy of a continuous map that removes its nperiodic orbits) can be reduced to equivariant fixedpoint problems. This answers a conjecture of Klein and Williams, and allows us to interpret their invariant as a class in topological restriction homology (TR), coinciding with a class defined earlier in the thesis of Iwashita and separately by Luck. This is joint work with Kate Ponto.
Arpon Raksit – TBA
TBA
Calista Bernard – TBA
TBA
Martina Rovelli – TBA
TBA
Sune Precht Reeh – TBA
TBA