Hi, I am Klaus Mattis, a second year PhD-student at JGU Mainz, under the supervision of Tom Bachmann.

For more information, consult my CV.

You can reach me via mail at klaus.mattis@uni-mainz.de

Research Interests

My research interests are

• Unstable and stable motivic homotopy theory
• Higher categories, particularly higher topoi
• Algebraic \(K\)-theory

Work

• Unstable \(p\)-completion in motivic homotopy theory (Arxiv) (PDF)

  We define unstable \(p\)-completion in general \(\infty\)-topoi and the unstable motivic homotopy category, and prove that the \(p\)-completion of a nilpotent sheaf or motivic space can be computed on its Postnikov tower. We then show that the (\(p\)-completed) homotopy groups of the \(p\)-completion of a nilpotent motivic space \(X\) fit into short exact sequences
  \( 0 \to \mathbb L_0 \pi_n(X) \to \pi_n^p(X^{\wedge}_p) \to \mathbb L_1 \pi_{n-1}(X) \to 0\),
  where the \(\mathbb L_i\) are (versions of) the derived \(p\)-completion functors, analogous to the classical situation.

• The pro-Nisnevich topology (Arxiv) (PDF)

  We construct the pro-Nisnevich topology, an analog of the pro-étale topology. We then show that the Nisnevich \(\infty\)-topos embeds into the pro-Nisnevich \(\infty\)-topos, and that the pro-Nisnevich \(\infty\)-topos is locally of homotopy dimension \(0\).

• Unstable arithmetic fracture squares in \(\infty\)-topoi (Arxiv) (PDF)

  We show that for a large class of \(\infty\)-topoi there exist unstable arithmetic fracture squares, i.e. squares which recover a nilpotent sheaf \(F\) as the pullback of the rationalization of \(F\) with the product of the \(p\)-completions of \(F\) ranging over all primes \(p \in \mathbb Z\).

• The derived \(\infty\)-category of Cartier modules (joint with Timo Weiß) (Arxiv) (PDF)

  For an endofunctor \(F\colon\mathcal{C}\to\mathcal{C}\) on an (\(\infty\)-)category \(\mathcal{C}\) we define the \(\infty\)-category \(\operatorname{Cart}(\mathcal{C},F)\) of generalized Cartier modules as the lax equalizer of \(F\) and the identity. This generalizes the notion of Cartier modules on \(\mathbb{F}_p\)-schemes considered in the literature. We show that in favorable cases \(\operatorname{Cart}(\mathcal{C},F)\) is monadic over \(\mathcal{C}\). If \(\mathcal{A}\) is a Grothendieck abelian category and \(F\colon\mathcal{A}\to\mathcal{A}\) is an exact and colimit-preserving endofunctor, we use this fact to construct an equivalence \(\mathcal{D}(\operatorname{Cart}(\mathcal{A},F)) \simeq \operatorname{Cart}(\mathcal{D}(\mathcal{A}),\mathcal{D}(F))\) of stable \(\infty\)-categories. We use this equivalence to give a more conceptual construction of the perverse t-structure on \(\mathcal{D}^b_{\operatorname{coh}}(\operatorname{Cart}(\operatorname{QCoh}(X), F_*))\) for any Noetherian \(\mathbb{F}_p\)-scheme \(X\) with finite absolute Frobenius \(F\colon X\to X\).

Talks

• Unstable \(p\)-completion in motivic homotopy theory at YoungHom (Slides)
• Proof of the Hopkins-Morel-Hoyois Theorem at International Workshop on Algebraic Topology, Shanghai

Academic service

I am a co-organizer of the winter school on unstable motivic homotopy theory in Mainz.

Impressum

Angaben gem. § 5 TMG:
Klaus Mattis
Institut für Mathematik
Johannes Gutenberg-Universität
Staudingerweg 9
55128 Mainz