Hi, I am Klaus Mattis, a second year PhDstudent at JGU Mainz, under the supervision of Tom Bachmann.
For more information, consult my CV.
You can reach me via mail at klaus.mattis@unimainz.de
Research Interests
My research interests are Unstable and stable motivic homotopy theory
 Higher categories, particularly higher topoi
 Algebraic \(K\)theory
Work

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
\( 0 \to \mathbb L_0 \pi_n(X) \to \pi_n^p(X^{\wedge}_p) \to \mathbb L_1 \pi_{n1}(X) \to 0\),
where the \(\mathbb L_i\) are (versions of) the derived \(p\)completion functors, analogous to the classical situation. 
We construct the proNisnevich topology, an analog of the proétale topology. We then show that the Nisnevich \(\infty\)topos embeds into the proNisnevich \(\infty\)topos, and that the proNisnevich \(\infty\)topos is locally of homotopy dimension \(0\).

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 \(p \in \mathbb Z\).

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 colimitpreserving 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 tstructure 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 HopkinsMorelHoyois Theorem at International Workshop on Algebraic Topology, Shanghai
Academic service
I am a coorganizer of the winter school on unstable motivic homotopy theory in Mainz.Impressum
Angaben gem. § 5 TMG:Klaus Mattis
Institut für Mathematik
Johannes GutenbergUniversität
Staudingerweg 9
55128 Mainz