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
During the summer term 26 I will offer the lecture course Geometrie und Topologie for Bachelor students and the seminar Funktionentheorie (based on last term's lecture) 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) - 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
to appear in Forum of Mathematics Pi (100 pages, arXiv: 2103.13911) - 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
to appear in Annals of Mathematics (63 pages, arXiv: 2009.07225) - 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