Hermitian K-theory and Topological Hochschild Homology

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.