December 9-11, 2024, Venice, Italy

This three days workshop aims at gathering experts and younger researchers active in the research areas of real topological Hochschild homology, Grothendieck-Witt theory and algebraic K-theory.

The talks will take place in the Palazzo Giustinian LolinLink opens in a new window, on the Canal Grande, site of the Venice Warwick campus.

Speakers:

The participants will be hosted at the Foresteria LeviLink opens in a new window. For questions, please contact the organiser at .

The workshop is kindly supported by the EPSRC grant: "Characteristic polynomials for symmetric forms" (EP/W019620/1).

Workshop Programme

Abstracts

Gabriel Angelini-Knoll: Real syntomic cohomology

Park and Hornbostel construct the syntomic cohomology of certain rings with involution as the associated graded of a filtration on Real topological cyclic homology. This refines the construction of syntomic cohomology due to Bhatt-Morrow-Scholze. In my talk, I will discuss an extension of this theory to the setting of ring spectra with involution, which refines the even filtration of Hahn-Raksit-Wilson. This theory is closely related to Artin-Tate real motivic cohomology and it allows one to study Lichtenbaum-Quillen statements for ring spectra with involution. This talk is based on work in progress with H. J. Kong and J. D. Quigley.

Dustin Clausen: Ausoni-Rognes' Galois descent conjecture

Ausoni-Rognes conjectured that T(n+1)-local K-theory has Galois descent for T(n)-local G-Galois extensions, with G a finite group. In joint work with Akhil Mathew, Niko Naumann, and Justin Noel, we proved this when G is a p-group, and reduced the general case to the case of a cyclic group of prime-to-p order. I will describe recent work with Robert Burklund where we handle that remaining case. The methods are, by necessity, quite different from those of [CMNN], as I will explain in the talk.

Thomas Nikolaus: Real TC and L-theory

We will review the theory of real cyclotomic spectra. The first main result will be that the geometric fixed points of real TC only depend on the geometric fixed points of real THH. As a result we deduce an easy formula for real TC and can show that it is equivalent to normal L-Theory (for any Poincarécategory). If time permits, we will explain how this relates to the theory of real topological Cartier modules.

Irakli Patchkoria: On the K(n)-local duality for infinite groups

This is joint work with Gijs Heuts. Given a discrete group G with a finite classifying space for proper actions, we give a new description of Klein’s dualising complex D_G in terms of proper equivariant stable homotopy theory. This allows us to generalise the K(n)-local ambidexterity result for finite groups to infinite groups: For a discrete group G with a finite classifying space for proper actions and any K(n)-local spectrum E with a G-action, we show that the norm map (E \otimes D_G)_{hG} \to E^{hG} is a K(n)-local equivalence. This equivalently means that the Farrell-Tate construction E^{tG} vanishes K(n)-locally. We relate this result to the character theory and compute a rational localisation of E^{tG} for Morava E-theory and various groups such as certain arithmetic groups, mapping class groups and Out(F_n)-s. We will also mention an application in the n=1 case which is joint work with Naomi Andrew. In this case we get that E^{tG} is rational and one can for example fully compute the p-adic Farrell-Tate K-theory of Out(F_{p+1}).

Maxime Ramzi: An ahistorical approach to the DGM theorem (Subtitle: towards real DGM?)

I will discuss a somewhat categorical proof of the celebrated Dundas-Goodwillie-McCarthy theorem, reviewing along the way the Dundas-McCarthy and Lindenstrauss-McCarthy theorems from the perspective of trace theories à la Kaledin-Nikolaus.

Part of the point is to make the relevant results sufficiently abstract/formal that the proofs can carry over to the real setting, though I will not discuss that in detail.

This talk will be based on work that has a strong overlap with work of Harpaz, Nikolaus and Saunier.

Jan Steinebrunner: Integral characteristic classes for Poincaré duality spaces

A surprising result of Berglund and Madsen says that for d>2 and as g -> infinity the rational cohomology of BhAut( (S^d x S^d)^{#g} ) is a free graded commutative algebra on classes indexed by the cohomology of Out(F_n). Using joint work in progress with Shaul Barkan, I will explain how to arrive at these classes from a categorical perspective. This in particular yields integral refinements in BhAut(M) for any oriented Poincaré duality space M.

We use that C^*(M) is a (twisted) E-infinity-Frobenius algebra: it has both an E-infinity-algebra structure and a "trace" [M]: C^*(M) --> Z[n], which together exhibit a (shifted) self-duality of C^*(M). Via the theory of cyclic and modular infinity-operads we show that the graph bordism category (as studied by Galatius) contains the universal example of an E-infinity-Frobenius algebra. The mapping spaces of this category contain a copy BOut(F_n) and so this space yields universal operations associated to any family of Poincaré duality spaces.

Adela Zhang: An equivariant Adams spectral sequence for tmf(2)

In this talk, I will explain how to compute the -equivariant relative Adams spectral sequence for the Borelification of tmf(2). This yields an entirely algebraic computation of the 3-local homotopy groups of tmf. The final answer is well-known of course — the novelty here is that the rASS is completely determined by its E_1-page as a cochain complex of Mackey functors. Explicitly, the input consists of the Hopf algebroid structure on modulo transfer, which is deduced from the structure maps on the equivariant dual Steenrod algebra, as well as the knowledge of (tmf(2)) along with the -action. Then we construct a bifiltration on tmf(2) and use synthetic arguments to deduce the Adams differentials from the associated square of spectral sequences. The rASS degenerates on for tridegree reasons and stabilizes to a periodic pattern that essentially lies within a band of slope 1/4. This is joint work with Jeremy Hahn, Andrew Senger, and Foling Zou.