Personal homepage of Ferdinand Wagner

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.

Current research

I'm trying to construct a version of \(q\)-de Rham cohomology that already lives over 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 it has become apparent that this should be related to a certain refinement \(\mathrm{THH}^{\mathrm{ref}}(-)\) of topological Hochschild homology, constructed by Peter Scholze and Alexander Efimov. More precisely, \(\mathrm{THH}^{\mathrm{ref}}(-/\mathrm{KU})\) should give rise to such a Habiro-valued cohomology theory, at least for varieties over \(\mathbb Q\), but also in some integral situations. I would like to understand this construction better and I'm curious to find out what happens over higher chromatic bases.

Furthermore, there seems to be a connection to knot theory and the quantum modularity conjectures of Garoufalidis and Zagier. In particular, the Habiro-valued cohomology theory recovers their generalised Habiro rings when evaluated on étale algebras over \(\mathbb Z\). I would also like to understand more about this mysterious connection.

Preprints

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

1. \(q\)-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.
2. Derived \(q\)-Hodge complexes and refined \(\mathrm{TC}^-\) (joint with Samuel Meyer).
   
   

   The analytic spectrum of \(\pi_0\mathrm{TC}^-((\mathrm{KU}_p^\wedge\otimes\mathbb Q)/\mathrm{KU}_p^\wedge)\).
   We show that in a surprising number of cases the \(q\)-de Rham complex can be equipped with a functorial filtration that \(q\)-deforms the usual Hodge filtration. In these situations, we can construct a fully functorial “derived \(q\)-Hodge complex” with all expected properties, despite the no-go result in general. Using these complexes (and the ideas that go into their construction), we compute \(\pi_*\mathrm{TC}^{-,\mathrm{ref}}(\mathrm{ku}\otimes\mathbb Q/\mathrm{ku})\) and \(\pi_*\mathrm{TC}^{-,\mathrm{ref}}(\mathrm{KU}\otimes\mathbb Q/\mathrm{KU})\). This paper builds heavily on the connection between \(q\)-de Rham cohomology and \(\mathrm{ku}\) that was discovered by Devalapurkar and Raksit.
3. \(q\)-Witt vectors and generalised Habiro rings (in preparation).
   We show that in all cases where the \(q\)-de Rham complex can be equipped with a \(q\)-deformation of the Hodge filtration, the derived \(q\)-Hodge complex functorially descends to the Habiro ring. In the special case of étale extensions of \(\mathbb Z\), the descended object recovers the generalised Habiro ring defined by Garoufalidis–Scholze–Zagier. We also show that the Habiro-descent can be obtained via genuine equivariant homotopy theory: Roughly, derived \(q\)-Hodge complexes arise from the \(S^1\)-action on \(\mathrm{THH}(-/\mathrm{KU})\), the Habiro descent comes from making the action of the finite subgroups \(C_m\subseteq S^1\) genuine.

Talks

1. Refined \(\mathrm{TC}^-\) over \(\mathrm{ku}\) and derived \(q\)-Hodge complexes.
   Workshop on Dualisable Categories & Continuous K-Theory, Bonn 2024.

Notes

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 Derived \(q\)-Hodge complexes and refined \(\mathrm{TC}^-\).
2. 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.
3. 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.