Personal homepage of Ferdinand Wagner

I'm currently a PhD student at the University of Bonn/MPI 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.

Doctoral thesis

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 \(\mathbb Z[[q-1]]\). Interestingly, this seems to have a connection to knot theory and the quantum modularity conjectures of Garoufalidis and Zagier. In particular, the desired cohomology theory should recover their generalised Habiro rings when evaluated on étale \(\mathbb Z\)-algebras. I would also like to understand more about this mysterious connection.

Previous writings

• My master thesis: I analysed a modification of \(q\)-de Rham cohomology that was hoped to descend to the Habiro ring \(\mathcal H\). Unfortunately, I proved that this modification can not even be functorial over \(\mathbb Z[[q-1]]\). Still, this led to some interesting constructions (\(q\)-Witt vectors and \(q\)-de Rham-Witt complexes) that are expected to show up in the true picture.
• \(K\)-Theory notes: Notes for Fabian Hebestreit's lecture on \(\infty\)-categories and \(K\)-Theory, held at the University of Bonn in the winter term 2021/22.