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.
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 two further members:
- Julius Frank (Post-Doc)
- Vikram Nadig (Doctoral student)
- Jan Geschwinde (Master student)
Teaching
During the summer term 25 I will only teach Higher categories and algebraic K-theory III (thursdays 12-2pm, V5-230) as I am on sabbatical.
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.
Publications and Preprints
Commutative algebra
- with P.Scholze: A note on higher almost ring theory.
arXiv: 2409.01940 (9 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.
to appear in Transactions of the American Mathematical Society (41 pages, 2021, arXiv: 2111.00069) - 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, vol. 154, no. 6 (2024), pp. 1749 - 1785 - with W.Steimle: Stable moduli spaces of hermitian forms.
arXiv: 2103.13911 (with an appendix by Y.Harpaz, 94 pages, 2021, in revision) - 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.
arXiv: 2009.07225 (63 pages, 2020, in revision) - 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.
to appear in Acta Mathematica (149 pages, 2020, arXiv: 2009.07224) - 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