Klaus Mattis
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
Motivic homotopy theory is a branch of algebraic geometry and algebraic topology that studies schemes with tools inspired by classical homotopy theory. It blends techniques from both geometry and topology to study the so-called "motives" of varieties.
For more details, see Wikipedia: Motivic Homotopy Theory .
Higher category theory generalizes the notion of categories, functors, and natural transformations to "n-categories," capturing deeper homotopy-theoretic information. An (∞,1)-topos (or higher topos) is a higher-categorical version of a topos, providing a unifying framework for homotopy theories of various geometric contexts.
For more details, see Wikipedia: Topos (classical notion) and nLab: (∞,1)-Topos .
Algebraic K-theory is a tool used in both geometry and topology to study projective modules over a ring, vector bundles on a scheme, and more. It is closely tied to areas like motivic homotopy theory and is fundamental in exploring deep connections between number theory, geometry, and topology.
For more details, see Wikipedia: 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 \(p \in \mathbb{Z}\).
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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.I am an organizer of the GAUS Junior AG on Maps between spherical group rings in Mainz.
Impressum
Angaben gem. § 5 TMG:Klaus Mattis
Institut für Mathematik
Johannes Gutenberg-Universität
Staudingerweg 9
55128 Mainz