34th NRW Topology Meeting - Abstracts

Maxime Ramzi (Münster): The dualizable Brauer group

Brauer groups are classically rich invariants of schemes, that give a geometric description of some étale cohomology groups. In this talk, I will explain how the definition naturally extends to homotopy theory, and how it leads to the "dualizable" or "categorical" Brauer group. The goal is to say a few nontrivial things about it, and discuss the question of whether it agrees with the classical Brauer group. Parts of the talk report some work in progress with Ben Antieau.


Julius Frank (Bielefeld): On DGAs with polynomial homology

It is surprisingly hard to find non-trivial examples of DGAs whose homology is polynomial in one generator. I will discuss one such example, which is a derived quotient of a discrete ring by an odd prime. I will also give one reason why it is hard to find examples: If the homology of a DGA A is polynomial over a perfect F_p-algebra and the underlying ring spectrum of A refines to an E_3-ring spectrum, then A must already be trivial in a certain sense.


Thor Wittich (Osnabrück): From Operations on Milnor-Witt K-theory to Motivic Knot Invariants

Understanding (co-)homology operations has shown to be very useful in algebraic topology and algebraic geometry. In this talk we focus on operations of Milnor-Witt K-theory, where the latter is an invariant of smooth algebraic varieties which arises naturally in motivic homotopy theory. The structure of the talk is as follows. We give a short introduction to Milnor-Witt K-theory. Afterwards we explain results on additive, stable and non-additive operations on Milnor-Witt K-theory. Finally, we indicate how these operations lead to motivic knot invariants. The last part is ongoing joint work with Matthias Wendt based on unpublished ideas of Aravind Asok and Matthias Wendt.


Elizabeth Tatum (Bonn): Applications of equivariant Brown-Gitler Spectra

In the 1980s, Mahowald and Kane used integral Brown-Gitler spectra to construct splittings of the cooperations algebras for ko, connective real k-theory, and ku, connective complex k-theory. These splittings helped make it feasible to do computations using the ko- and ku-based Adams spectral sequences. In recent work, Guchuan Li, Sarah Petersen, and I have constructed models for C_2-equivariant analogues of the integral Brown-Gitler spectra. In this talk, I will report on our progress towards using these spectra to construct C_2-equivariant analogues of these splittings, and related results.


Tom Bachmann (Mainz): E_oo-coalgebras and p-adic homotopy theory

We prove that taking "chains with coefficients in an separably closed field k of characteristic p" induces a fully faithful functor from p-complete, nilpotent spaces to E_∞-coalgebras over k. This removes finiteness assumptions from a theorem of Mandell (jt. with Robert Burklund).


Jens Hornbostel (Wuppertal): Real topological Hochschild homology of perfectoid rings

We recall the definition of classical THH and its real refinement THR for rings and schemes with involution. Then we discuss some general results as well as some recent computations, in particular about perfectoid rings. The latter refines results of Bhatt-Morrow-Scholze on THH. Along the way, we establish a real refinement of the Hochschild-Kostant-Rosenberg theorem. This is joint work with Doosung Park.


Konrad Bals (Münster): The topological Cartier-Raynauld ring

Topological invariants around THH are strongly intertwined with the study of arithmetic cohomology theories. The category of p-typical topological Cartier modules TCart as defined by Antieau--Nikolaus gives a categorical framework for the comparison between Hesselholt--Madsens topological restriction homology and Bloch--Deligne--Illusies algebraic de-Rham--Witt complex. Writing TCart as a module category defines the topological Cartier--Raynaud ring and in this talk I will show how to compute its homotopy groups. In particular, after basechanging to Z_p this computation recovers the classical Cartier--Raynaud ring which acts on the de-Rham--Witt complex.


Kaif Hilman (MPIM): Equivariant Poincare duality and fixed points methods

In this talk, I will introduce the notion of Poincare duality spaces in the equivariant setting for compact Lie groups and discuss an approach relating equivariant Poincare duality with Poincare duality on the fixed points. I will then indicate several geometric phenomena that become visible from this purely homotopical theory. This is joint work with Dominik Kirstein and Christian Kremer.