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 and 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
- with P.Scholze: A note on higher almost ring theory.
arXiv: 2409.01940 (13 pages, 2024) - with A.Krause & M.Ramzi: A note on quadratic forms.
Bulletin of the London Mathematical Society, vol. 56, no. 5 (2024), pp. 1803-1818
Algebraic & geometric topology
- with M.Land, M.Weiss & C.Winges: Homology manifolds and euclidean bundles.
arXiv: 2406.14677 (11 pages, 2024, in revision) - with I.Bokor, D.Crowley, S.Friedl, D.Kasprowski, M.Land & J.Nicholson: Connected sum decompositions of high-dimensional manifolds.
Matrix Annals, vol. 4 (2021), pp. 5-30 - with S.Sagave: Homotopical and operator algebraic twisted K-theory.
Mathematische Annalen, vol. 378 (2020), pp. 1021-1059 - with C.Schlichtkrull & S.Sagave: Multiplicative parametrized homotopy theory via symmetric spectra in retractive spaces.
Forum of Mathematics Sigma, vol. 8 (2020), article no. e16, 84 pages - with M.Land, W.Lück & O.Randal-Williams: A vanishing theorem for tautological classes of aspherical manifolds.
Geometry & Topology, vol. 25 (2021), pp. 47-110 - with N.Perlmutter: Cobordism categories and moduli spaces of odd dimensional manifolds.
Advances in Mathematics, vol. 353, no. 7 (2019), pp. 526-590 - with M.Joachim: Twisted Spin cobordism and positive scalar curvature.
Journal of Topology, vol. 13, no. 1 (2020), pp. 1-58 - with S.Büscher, O.Röndigs & M.Stelzer: The Goodwillie Spectral Sequence for iterated loopspaces and topological realisation at odd primes.
Algebraic and Geometric Topology, vol. 13, no. 1 (2013), pp. 127-169
Category theory
- with J.Steinebrunner: A short proof that Rezk's nerve is fully faithful.
International Mathematics Research Notices, vol. 2025, no.4 (2025), article no. rnaf021, 13 pages - with R.Haugseng, S.Linskens & J.Nuiten: Two-variable fibrations, factorisation systems and categories of spans.
Forum of Mathematics Sigma, vol. 11 (2023), article no. e111, 70 pages - with R.Haugseng, S.Linskens & J.Nuiten: Lax monoidal adjunctions, two-variable fibrations and the calculus of mates.
Proceedings of the London Mathematical Society, vol. 127, no. 4 (2023), pp. 889-957 - with G.Heuts & J.Ruit: A short proof of the straightening theorem.
Transactions of the American Mathematical Society B, vol. 12 (2025), pp. 697 - 747 - with A.Krause: Mapping spaces in coherent nerves.
arXiv: 2011.09345 (14 pages, 2020)
Algebraic & hermitian K-theory
- with A.Lachmann & W.Steimle: The localisation theorem for the K-theory of stable categories.
Proceedings of the Royal Society of Edinburgh A, vol. 154, no. 6 (2024), pp. 1749 - 1785 - with W.Steimle: Stable moduli spaces of hermitian forms.
to appear in Forum of Mathematics Pi (with an appendix by Y.Harpaz, 100 pages, arXiv: 2103.13911) - with B.Calmès, E.Dotto, Y.Harpaz, M.Land, K.Moi, D.Nardin, T.Nikolaus & W.Steimle: Hermitian K-Theory for stable ∞-categories III: Grothendieck-Witt groups of rings.
to appear in Annals of Mathematics (63 pages, arXiv: 2009.07225) - with B.Calmès, E.Dotto, Y.Harpaz, M.Land, K.Moi, D.Nardin, T.Nikolaus & W.Steimle: Hermitian K-Theory for stable ∞-categories II: Cobordism categories and Additivity.
Acta Mathematica, vol. 235, no. 2 (2025), pp. 149 - 400 - with B.Calmès, E.Dotto, Y.Harpaz, M.Land, K.Moi, D.Nardin, T.Nikolaus & W.Steimle: Hermitian K-Theory for stable ∞-categories I: Foundations.
Selecta Mathematica, vol. 29, no. 1 (2023), article no. 10, 269 pages - with M.Land & T.Nikolaus: On the homotopy type of L-spectra of the integers.
Journal of Topology, vol. 14, no. 1 (2021), pp. 183-214, Erratum
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