Ferdinand Wagner

Personal homepage of Ferdinand Wagner

Max-Planck Insitute for Mathematics
Vivatsgasse 7
53111 Bonn, Germany

Office 404

The obligatory Oberwolfach photo.

I'm currently a PhD student at the MPIM/University of Bonn, under the supervision of Peter Scholze. I'm interested in cohomology theories for arithmetic schemes, especially in the global (as opposed to \(p\)-adic) case. I'm also a huge fan of applying higher categorical and homotopical methods to algebra problems.

Here's a CV and my thesis.

Current research interests

Peter Scholze has raised the question whether there exists a version of \(q\)-de Rham cohomology with coefficients in the Habiro ring \[\mathcal H=\lim_{m\in\mathbb N}\mathbb Z[q]_{(q^m-1)}^\wedge\] rather than the power series ring \(\mathbb Z[[q-1]]\). Recently, Scholze has constructed an analytic Habiro stack, whose coherent cohomology gives rise to a cohomology theory with coefficients in an analytic version of the Habiro ring. In my thesis, I construct an algebraic version of Habiro cohomology, with coefficients in the completion of \(\mathcal H\bigl[\frac 1N\bigr]\) for some large enough integer \(N\), and I'd like to find out how this relates to Scholze's construction.

Efimov and Scholze have constructed certain refinements \(\mathrm{THH}^{\mathrm{ref}}\) and \(\mathrm{TC}^{-,\mathrm{ref}}\) of topological Hochschild and negative cyclic homology. Using an approprately defined even filtration on \(\mathrm{THH}^{\mathrm{ref}}(-/\mathrm{KU})\), it should be possible to construct another version of analytic Habiro cohomology, at least for varieties over \(\mathbb Q\). I aim to reconcile this with Scholze's construction and I'm curious to find out what happens over higher chromatic bases.

Habiro cohomology seems to be related to 3-manifold invariants and the quantum modularity conjectures of Garoufalidis and Zagier. For example, algebraic Habiro cohomology recovers their generalised Habiro rings when evaluated on étale algebras over \(\mathbb Z\); moreover Garoufalidis–Wheeler have recently found constructions of explicit classes in algebraic Habiro cohomology using perturbative Chern–Simons theory. I would like to understand more about this mysterious connection.

Preprints

My preprints can also be found on the arXiv. The versions here are optimised for (or rather against) badboxes and my typographical taste.

Peter Scholze has raised the question whether some variant of the \(q\)-de Rham complex is already defined over the Habiro ring. Such a variant should then be called (algebraic) Habiro cohomology.

We show that algebraic Habiro cohomology exists whenever the \(q\)-de Rham complex can be equipped with a \(q\)-Hodge filtration: a \(q\)-deformation of the Hodge filtration, subject to some reasonable conditions. To any such \(q\)-Hodge filtration we'll associate a small modification of the \(q\)-de Rham complex, which we call the \(q\)-Hodge complex, and show that it descends canonically to the Habiro ring. This construction recovers and generalises the Habiro ring of a number field of Garoufalidis–Scholze–Wheeler–Zagier and is closely related to the \(q\)-de Rham--Witt complexes from my previous work. Conjecturally there should also be a close connection to Scholze's analytic Habiro stack.
Work of Sanath Devalapurkar and Arpon Raksit suggests a relation between \(q\)-de Rham cohomology and \(\mathrm{THH}(-/\mathrm{ku})\). In this article we work out this relation: If \(R\) is quasi-syntomic with a spherical \(\mathbb E_2\)-lift \(\mathbb S_R\), then the graded pieces of the even filtration on \(\mathrm{TC}^-(\mathrm{ku}\otimes\mathbb S_R/\mathrm{ku})\) are given by the completion of a certain \(q\)-Hodge filtration on the derived \(q\)-de Rham complex of \(R\). We also explain how to obtain the \(q\)-Hodge complex and its Habiro descent from \(\mathrm{THH}(\mathrm{KU}\otimes\mathbb S_R/\mathrm{KU})\) and its genuine equivariant structure.
As a consequence of Efimov's proof of rigidity of the \(\infty\)-category of localising motives, Efimov and Scholze have constructed refinements of localising invariants such as \(\mathrm{THH}\) and \(\mathrm{TC}^-\). These refinements often contain vastly more information than the original invariant. In this article we explain a general recipe how to compute the refinements in certain situations. We then apply this recipe to compute \(\mathrm{TC}^{\mathrm{ref},-}(\mathrm{ku}\otimes\mathbb{Q}/\mathrm{ku})\) and \(\mathrm{TC}^{\mathrm{ref},-}(\mathrm{KU}\otimes\mathbb{Q}/\mathrm{KU})\). The result has a rather surprising geometric description (see the picture) and contains non-trivial information modulo any prime, in contrast to the unrefined \(\mathrm{TC}^-\).

The analytic spectrum of \(\pi_0\mathrm{TC}^-((\mathrm{KU}_p^\wedge\otimes\mathbb Q)/\mathrm{KU}_p^\wedge)\).

This paper builds heavily on the connection between \(q\)-de Rham cohomology and topological Hochschild homology over \(\mathrm{ku}\) that was discovered by Devalapurkar and Raksit and explored in my eponymous article.
We introduce a “\(q\)-version” of Witt vectors and de Rham–Witt complexes and show that they are closely related to a variant of \(q\)-de Rham cohomology, which we call the “\(q\)-Hodge complex”. We also show an unfortunate no-go result for coordinate independence of the \(q\)-Hodge complex.

Miscellaneous

Slides for a talk at the workshop on Dualisable Categories & Continuous K-Theory in Bonn 2024.
In this short note we explain how the \(q\)-de Rham for smooth algebras over \(\mathbb Z\) can be glued together from its \(p\)-completions (defined in terms of Bhatt–Scholze's \(q\)-crystalline cohomology) and its rationalisation (defined as a base change of de Rham cohomology). This is mostly formal, but not completely trivial. Now incorporated into my thesis.
Notes for Fabian Hebestreit's lecture on \(\infty\)-categories and \(K\)-Theory, held at the University of Bonn in the winter term 2021/22.
Contains a first definition of \(q\)-Witt vectors and \(q\)-de Rham–Witt complexes as well as the no-go result for the \(q\)-Hodge complex. These ideas are developed more systematically in my paper on \(q\)-Witt vectors and \(q\)-Hodge complexes.