Personal homepage of Ferdinand Wagner

Current research interests

• Habiro cohomology\(\).

  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 a version of such a theory with coefficients in an analytic variant 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 \(N\). I'd like to find out how this relates to Scholze's analytic construction.

• Refined \(\mathrm{THH}\) and \(\mathrm{TC}^-\).

  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 recover Scholze's analytic Habiro cohomology, along with a modification at roots of unity (which fixes an issue with the current construction). I aim to work this out and I'm curious to find out what happens over higher chromatic bases.

• Connections to 3-manifold topology\(\).

  Habiro cohomology seems to have a deep connection to 3-manifold invariants and the quantum modularity conjectures of Garoufalidis and Zagier. In the case of étale algebras over \(\mathbb Z\), this is the content of Garoufalidis–Scholze–Wheeler–Zagier's work on the Habiro ring of a number field (combined with the fact that algebraic Habiro cohomology recovers their construction). But the story seems to continue beyond the étale case. For example, 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 updated more frequently and optimised for (or rather against) badboxes and for (not against) my typographical taste.

1. \(q\)-Hodge complexes over the Habiro ring.
   This paper studies (algebraic) Habiro cohomology: a variant 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 \(\mathbb Z[[q-1]]\).

   We show that algebraic Habiro cohomology can be constructed 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 cohomology.

   While there (provably!) exists no functorial \(q\)-Hodge filtration for all smooth schemes \(X\) over \(\mathbb Z\), we show that a functorial \(q\)-Hodge filtration does exist as soon as all primes \(\leqslant \mathrm{dim}(X/\mathbb Z)\) become inverted. Thus, we obtain a reasonably well-behaved theory of Habiro cohomology “away from small primes”.
2. \(q\)-de Rham cohomology and topological Hochschild homology over \(\mathrm{ku}\).
   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 such a 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 (a.k.a. algebraic Habiro cohomology) from \(\mathrm{THH}(\mathrm{KU}\otimes\mathbb S_R/\mathrm{KU})\) and its genuine equivariant structure.
3. \(q\)-Hodge complexes and refined \(\mathrm{TC}^-\)
   (joint with Samuel Meyer).
   As a consequence of Efimov's theorem on the rigidity of the \(\infty\)-category of localising motives, Efimov and Scholze have constructed refinements of localising invariants such as \(\operatorname{THH}\) 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 \(\pi_0\mathrm{TC}^{-,\mathrm{ref}}((\mathrm{KU}_p^\wedge\otimes\mathbb Q)/\mathrm{KU}_p^\wedge)\).

   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.
4. \(q\)-Witt vectors and \(q\)-Hodge complexes.
   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.

Slides from talks

1. \(q\)-Hodge filtrations, Habiro cohomology, and \(\mathrm{ku}\).
   My thesis defense talk.
2. Refined \(\mathrm{TC}^-\) over \(\mathrm{ku}\) and derived \(q\)-Hodge complexes.
   Contributed talk at the workshop on Dualisable Categories & Continuous \(K\)-Theory in Bonn 2024.

Miscellaneous

1. The global \(q\)-de Rham complex.
   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.
2. \(\infty\)-Categories in topology.
   Notes from a QED Academy course 2023. Roughly 70% finished.
3. Algebraic and Hermitian \(K\)-Theory.
   Notes for Fabian Hebestreit's lecture on \(\infty\)-categories and \(K\)-Theory, held at the University of Bonn in the winter term 2021/22.
4. My master thesis\(\).
   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.