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
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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. -
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 \(0\).
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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 \(p \in \mathbb Z\).
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