Hermitian K-theory and Topological Hochschild Homology
Description
This workshop is organised around the themes of Hermitian K-theory and Topological Hochschild Homology with the goal of introducing a wider audience to the recent developments in Hermitian K-theory through Poincaré categories while supplementing this with a survey on general approaches to computing algebraic K-theory through trace methods and Topological Hochschild Homology.
Invited Speakers
Emanuele Dotto (University of Warwick, UK)
Steffen Sagave (Radboud University Nijmegen, The Netherlands)
Time and Location
The workshop will take place from Monday 20 to Wednesday 22 March 2023 at Radboud University Nijmegen, The Netherlands. The workshop begins on Monday morning and concludes at lunch time on Wednesday. All talks will be in HG00.303.
Schedule
All talks will be in HG00.303.
Mini Courses
Emanuele Dotto: Hermitian K-theory for stable infinity-categories
Steffen Sagave: Topological Hochschild Homology
Contributed Talks
Herman Rohrbach: A completion theorem for Hermitian K-theory
The Atiyah-Segal completion theorem (1969) gives a way of approximating, for a compact Lie group G, the G-equivariant complex topological K-theory of a compact G-space with the non-equivariant K-theory of an auxiliary topological space, which is closely related to the classifying space of G. An analogue of this theorem for algebraic K-theory has been proven by Amalendu Krishna (2018), but as of yet no such analogue exists for Hermitian K-theory. In this talk, we will prove the basic case of the completion theorem for the equivariant Hermitian K-theory of a scheme with a trivial torus action using the theory of derived completion.
Simon Pepin Lehalleur: Quadratic conductor formulas and Hermitian K-theory
Classical enumerative geometry often involves identities between coherent and topological (or motivic) invariants. For instance, the Deligne-Milnor conductor formula expresses the Euler characteristic of the vanishing cycles at an isolated hypersurface singularity in terms of the Jacobi algebra of the singularity. In quadratic enumerative geometry, numerical invariants are refined into classes in the Grothendieck-Witt ring of the base field, using Chow-Witt groups and Hermitian K-theory. I will explain a joint work with Marc Levine and Vasudevan Srinivas where we proved a quadratic conductor formula in a very special case and discuss what could be expected for more general singularities.
Andrei Konovalov: Topological K-theory of dg-categories and the lattice conjecture
I will discuss the problem of constructing a natural rational structure on periodic cyclic homology of dg-algebras and dg-categories. The promising candidate is A. Blanc’s topological K-theory, which is, roughly speaking, built from algebraic K-theory spectra of twists of a given dg-category. I will discuss structural properties and possible applications of Blanc’s topological K-theory and will show that it, indeed, provides a rational structure in a number of cases.
Jonas McCandless: Equivariant homotopy theory with infinite sums of transfer maps and TR
I will explain a new formalism for equivariant homotopy theory for infinite groups such as the integers which additionally encodes that certain infinite sums of transfer maps converge. I will discuss the relation to TR and the Witt vectors with coefficients. This is joint work with Achim Krause and Thomas Nikolaus.
Miguel Barrero: Globally equivariant commutativity for abelian groups
Global transfer systems parametrize different levels of commutativity in global homotopy theory, encoded by global $N_\infty$-operads. In this talk I will show how to compute all possible global transfer systems for abelian compact Lie groups.
Victor Saunier: The universal property of THH and the Dundas-McCarthy theorem
In joint work with Yonatan Harpaz and Thomas Nikolaus, we show that THH acquires a universal property when extended to bimodules of stable ∞-categories. We are able to compare this property with the universal property of K-theory defined in the same context and deduce a categorical version of the Dundas-McCarthy theorem, identifying the derivative of K-theory of square-zero extensions with THH.
Dominik Kirstein: A Twisted Bass-Heller-Swan Decomposition for Localising Invariants
Classically, the Bass-Heller-Swan decomposition relates the algebraic K-theory of a Laurent polynomial ring to the algebraic K-theory of its coefficient ring. Throughout the years, there have been many generalisations of this result and a version for localising invariants of stable infinity-categories has recently been proven by Saunier.
In this talk, I will explain how to obtain a twisted version of such a splitting for localising invariants if the stable infinity-category comes with an automorphism. As an application, one gets a splitting result for Waldhausen's A-theory of mapping tori. This is joint work with Christian Kremer.
Jack Davies: Functoriality of elliptic cohomology
The first elliptic cohomology theories appeared in the 1980‘s as multiplicative cohomology theories associated to an elliptic curve. Originally such constructions were functorial in the stable homotopy category, but refinements due to Goerss—Hopkins—Miller and Lurie lift this functoriality to one of infinity-categories. In this talk, we will discuss a further generalisation of these ideas and how they lead to the construction of, and relationships between, stable operations on elliptic cohomology theories.
Practical Information
Location
All talks will take place in Huygensgebouw in room HG00.303. More information on the location and how to get there can be found here: https://www.ru.nl/science/about-the-faculty/contact/how-get/
Public Transport
Regional public transport in the Netherlands is currently in a period of protracted strike action. Unfortunately Monday 20th and Wednesday 22nd March are both strike days. As such, you should leave extra time when trying to get anywhere (e.g., from the hotel to the conference location). Typically there are some buses still running, but the route is also walkable. More information on the strikes can be found here: https://9292.l/en/news/news-items/overview-calamities.
Coffee breaks
Coffee breaks at the workshop will be unconventional. Rather than coffee from a communal machine, we will instead be giving all participants a number of vouchers for the cafe in the Huygensgebouw. In an effort to help protect the environment and make this conference as "green" as possible, we strongly encourage you to bring your own reusable cup, if you have one.
Organisers
Eva Höning (Radboud University Nijmegen)
Magdalena Kedziorek (Radboud University Nijmegen)
Niall Taggart (Utrecht University)
Sponsors
Nederlandse Organisatie voor Wetenschappelijk Onderzoek (Dutch Research Council)
The K-theory foundation, Inc.
Radboud Excellence Initiative.