I am a professor of mathematics at Bielefeld University. I obtained my doctorate at the WWU Münster in 2014 and my habilitation at the RFWU Bonn in 2022, and was a lecturer at the University of Aberdeen before coming to Bielefeld in 2024.
Contact Information
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Research interests
Algebraic & geometric topology, homotopical algebra and more specifically: The homotopy theory of diffeomorphism groups, manifold bundles and cobordism categories and their relation to arithmetic groups and algebraic K- and L-theory; higher categories as foundations for the above.
Organisation
I am an editor of the Nagoya Mathematical Journal.
I am a member of the TRR 358 'Integral structures in geometry and representation theory', joint between the Universities of Bielefeld and Paderborn. My workgroup currently has the following members:
- Julius Frank (Post-Doc)
- Vikram Nadig (Doctoral student)
- Victor Saunier (Post-Doc)
Teaching
In the winter term 26/27 I am offering the lecture course Geometrie und Topologie II for Bachelor students and the seminar Kompakte Hausdorffräume for Master students.
Some lecture notes on algebraic topology, higher categories and algebraic K-theory are here and links to the webpages of previous courses at the end of the page.
Finally, here is an ongoing book project on higher categories and algebraic K-theory.
Publications and Preprints
Commutative algebra
- Jordan-Chevalley decompositions over imperfect fields.
with M.Hoff & W.Hoffmann
arXiv: 2604.08740 (8 pages, 2026, submitted) - A note on higher almost ring theory.
with P.Scholze
arXiv: 2409.01940 (13 pages, 2024) - A note on quadratic forms.
with A.Krause & M.Ramzi
Bulletin of the London Mathematical Society, vol. 56, no. 5 (2024), pp. 1803-1818
Algebraic & geometric topology
- Homology manifolds and euclidean bundles.
with M.Land, M.Weiss & C.Winges
arXiv: 2406.14677 (11 pages, 2024, in revision) - Connected sum decompositions of high-dimensional manifolds.
with I.Bokor, D.Crowley, S.Friedl, D.Kasprowski, M.Land & J.Nicholson
Matrix Annals, vol. 4 (2021), pp. 5-30 - Homotopical and operator algebraic twisted K-theory.
with S.Sagave
Mathematische Annalen, vol. 378 (2020), pp. 1021-1059 - Multiplicative parametrized homotopy theory via symmetric spectra in retractive spaces.
with C.Schlichtkrull & S.Sagave
Forum of Mathematics Sigma, vol. 8 (2020), article no. e16, 84 pages - A vanishing theorem for tautological classes of aspherical manifolds.
with M.Land, W.Lück & O.Randal-Williams
Geometry & Topology, vol. 25 (2021), pp. 47-110 - Cobordism categories and moduli spaces of odd dimensional manifolds.
with N.Perlmutter
Advances in Mathematics, vol. 353, no. 7 (2019), pp. 526-590 - Twisted Spin cobordism and positive scalar curvature.
with M.Joachim
Journal of Topology, vol. 13, no. 1 (2020), pp. 1-58 - The Goodwillie Spectral Sequence for iterated loopspaces and topological realisation at odd primes.
with S.Büscher, O.Röndigs & M.Stelzer
Algebraic and Geometric Topology, vol. 13, no. 1 (2013), pp. 127-169
Category theory
- A short proof that Rezk's nerve is fully faithful.
with J.Steinebrunner
International Mathematics Research Notices, vol. 2025, no.4 (2025), article no. rnaf021, 13 pages - Two-variable fibrations, factorisation systems and categories of spans.
with R.Haugseng, S.Linskens & J.Nuiten
Forum of Mathematics Sigma, vol. 11 (2023), article no. e111, 70 pages - Lax monoidal adjunctions, two-variable fibrations and the calculus of mates.
with R.Haugseng, S.Linskens & J.Nuiten
Proceedings of the London Mathematical Society, vol. 127, no. 4 (2023), pp. 889-957 - A short proof of the straightening theorem.
with G.Heuts & J.Ruit
Transactions of the American Mathematical Society B, vol. 12 (2025), pp. 697 - 747 - Mapping spaces in coherent nerves.
with A.Krause
arXiv: 2011.09345 (14 pages, 2020)
Algebraic & hermitian K-theory
- The localisation theorem for the K-theory of stable categories.
with A.Lachmann & W.Steimle
Proceedings of the Royal Society of Edinburgh A, vol. 154, no. 6 (2024), pp. 1749 - 1785 - Stable moduli spaces of hermitian forms.
with W.Steimle and an appendix by Y.Harpaz
Forum of Mathematics Pi, vol. 14 (2026), article no. e14, 112 pages - Hermitian K-Theory for stable ∞-categories III: Grothendieck-Witt groups of rings.
with B.Calmès, E.Dotto, Y.Harpaz, M.Land, K.Moi, D.Nardin, T.Nikolaus & W.Steimle
Annals of Mathematics, vol. 204, no. 1 (2026), pp. 119-220 - Hermitian K-Theory for stable ∞-categories II: Cobordism categories and Additivity.
with B.Calmès, E.Dotto, Y.Harpaz, M.Land, K.Moi, D.Nardin, T.Nikolaus & W.Steimle
Acta Mathematica, vol. 235, no. 2 (2025), pp. 149 - 400 - Hermitian K-Theory for stable ∞-categories I: Foundations.
with B.Calmès, E.Dotto, Y.Harpaz, M.Land, K.Moi, D.Nardin, T.Nikolaus & W.Steimle
Selecta Mathematica, vol. 29, no. 1 (2023), article no. 10, 269 pages - On the homotopy type of L-spectra of the integers. Erratum
with M.Land & T.Nikolaus
Journal of Topology, vol. 14, no. 1 (2021), pp. 183-214
Other writing
- Homology manifolds and Euclidean bundles.
Oberwolfach Reports vol. 21, no. 3, Workshop: Topologie (2024), pp. 1920-1922 - with M.Land: Apprenticeship with an algebraic surgeon.
Celebratio Mathematica, Ranicki memorial issue (2022) - with M.Land & T.Nikolaus: The initial quadratic form on an abelian group.
- Parametrised algebraic surgery.
Oberwolfach Reports vol. 17, no. 2-3, Workshop: Topologie (2020), pp. 1116-1121 - Algebraic Cobordism categories and Grothendieck-Witt theory.
Oberwolfach Reports vol. 17, no. 1, Workshop: Manifolds and Groups (2020), pp. 437-440 - The Tate Diagonal.
Oberwolfach Reports vol. 15, no. 2, Arbeitsgemeinschaft: Topological Cyclic Homology (2018), pp. 825-838 - Twisted Spin Cobordism.
Oberwolfach Reports vol. 11, no. 3, Workshop: Analysis, Geometry and Topology of positive scalar curvature (2014), pp. 2013-2014
Previous teaching pages
- Geometrie und Topologie (26)
- Funktionentheorie (25/26)
- Lineare Algebra II (24/25)
- Lineare Algebra I (24)