The schedule and details for the upcoming minicourse 'Trace Methods in Higher Algebra and Geometric Topology,' featuring speakers Maxime Ramzi and Florian Naef. Below are the schedule and the abstract provided by Maxime Ramzi:
Minicourse Schedule:
Monday, April 14 (Atlas Room)
9:30 – 10:30: Talk by Maxime Ramzi on "Goodwillie Calculus and Localizing Invariants"
10:30 – 10:55: Break
10:55 – 11:55: Talk by Florian Naef on "Dennis Trace in String Topology"
Tuesday, April 15 (Ruppert-116)
13:00 – 14:00: Talk by Florian Naef
14:00 – 14:25: Break
14:25 – 15:25: Talk by Maxime Ramzi
Wednesday, April 16 (MIN-2.01)
13:00 – 14:00: Talk by Florian Naef
14:00 – 14:25: Break
14:25 – 15:25: Talk by Maxime Ramzi
Thursday, April 17 (MIN-2.02)
9:30 – 10:30: Talk by Maxime Ramzi
10:30 – 10:50: Break
10:50 – 11:50: Talk by Florian Naef
Speaker: Florian Naef
Title: Dennis Trace in String Topology
Abstract: String topology can be thought of as the study of operations on the free loop space of a manifold. The operations are of the type cutting loops at intersections and regluing them in a different pattern. As the (supsension spectrum of the) free loop space can be identified with topological Hochschild homology of parametrized spectra, one can ask what these operations correspond to under this equivalence. Whereas part of the string topology structure can be explained by the fact that Poincare duality induces an equivalence between Hochschild homologoy and cohomology, this does not recover an operation called the loop coproduct. The loop coproduct arises from an invariant (the simple homotopy type) in algebraic K-theory by applying the Dennis trace.
I will try and give some background on the original geometric construction of the string topology operations, roughly following the historical discovery. We will then see various algebraic models and the connection to embedding calculus. And finally we will see how the Dennis trace appears.
Speaker: Maxime Ramzi
Title: The Goodwillie calculus of localizing invariants and the Dundas-Goodwillie-McCarthy theorem
The Dundas-Goodwillie-McCarthy theorem is a fundamental theorem in algebraic K-theory, the backbone of trace methods.
Recent advances in the theory of localizing invariants have helped clarify the proof of this theorem and related ingredients (the Dundas-McCarthy and Lindenstrauss-McCarthy theorems).
The goal of my lectures will be to present these recent advances and ultimately a proof of the DGM theorem. Roughly, an outline of the lectures is as follows :
1- Overview of the DGM theorem and proof strategy
2- Basics of the Goodwillie calculus of localizing invariants : first derivatives and trace theories
3- More on the Goodwillie calculus of localizing invariants : higher derivatives
4- Convergence, conclusion of the proof and leftovers from previous lectures.