Projects in Mathematics Muenster

Research area A: Arithmetic and Groups
Unit A1: Arithmetic, geometry and representations

Further Projects
CRC 1442: Geometry: Deformation and Rigidity - A04: New cohomology theories for arithmetic schemes We develop, study and compute certain global cohomology theories for schemes. These cohomology theories may be viewed as deformations of ‘classical’ cohomology theories over mixed characteristic or over the sphere spectrum. For example, de Rham–Witt cohomology is a deformation of de Rham cohomology over mixed characteristic. Topological Hochschild homology is a deformation of ordinary Hochschild homology over the sphere spectrum. The motivation for the project is to use these cohomology theories to attack deep problems in algebra, topology and arithmetic geometry. Our most ambitious application concerns zeta functions. online
CRC 1442: Geometry: Deformation and Rigidity - C02: Homological algebra for stable ∞-categories The goal of this project is to study the emerging area of homological algebra for stable infinity-categories. Concretely the major objective of this project is to study non-commutative motives as introduced by Blumberg–Gepner–Tabuada and the homotopy theory of chain complexes of stable infinity-categories that will be developed as part of the project, following up on pioneering work of Dyckerhoff. Moreover, we will explore the notion of a stable (infinity,infinity)-category and the corresponding higher version of spectra, which will besides its general importance also be relevant in setting up a rigid higher version of Quinn’s Ad-theories. online

Research Interests

Research Interests

$\bullet$ Homotopy theory and Higher Algebra.
$\bullet$ Algebraic $K$-theory.
$\bullet$ Field theories and mathematical Physics.
$\bullet$ (topological) Hochschild homology and non-commutative geometry.

Selected Publications

Selected Publications of Thomas Nikolaus

$\bullet$ T. Nikolaus and S. Sagave. Presentably symmetric monoidal $\infty$-categories are represented by symmetric monoidal model categories. Algebr. Geom. Topol., 17(5):3189–3212, 2017.

$\bullet$ D. Gepner, R. Haugseng, and T. Nikolaus. Lax colimits and free fibrations in $\infty$-categories. Doc. Math., 22:1225–1266, 2017.

$\bullet$ U. Bunke, T. Nikolaus, and M. Völkl. Differential cohomology theories as sheaves of spectra. J. Homotopy Relat. Struct., 11(1):1–66, 2016.

$\bullet$ D. Gepner, M. Groth, and T. Nikolaus. Universality of multiplicative infinite loop space machines. Algebr. Geom. Topol., 15(6):3107–3153, 2015.

$\bullet$ U. Bunke and T. Nikolaus. $T$-duality via gerby geometry and reductions. Rev. Math. Phys., 27(5):1550013, 46 pp., 2015.

$\bullet$ T. Nikolaus. Algebraic K-theory of $\infty$-operads. J. K-Theory, 14(3):614–641, 2014.

$\bullet$ T. Nikolaus and K. Waldorf. Lifting problems and transgression for non-abelian gerbes. Adv. Math., 242:50–79, 2013.

$\bullet$ T. Nikolaus, C. Sachse, and C. Wockel. A smooth model for the string group. Int. Math. Res. Not. IMRN, (16):3678–3721, 2013.

$\bullet$ T. Nikolaus and C. Schweigert. Equivariance in higher geometry. Adv. Math., 226(4):3367–3408, 2011.

$\bullet$ T. Nikolaus. Algebraic models for higher categories. Indag. Math. (N.S.), 21(1-2):52–75, 2011.

Current Publications

$\bullet $ Benjamin Antieau, Achim Krause, and Thomas Nikolaus. On the K-theory of $\mathbb Z/p^n$ – announcement. arXiv e-prints, April 2022. arXiv:2204.03420.

$\bullet $ Emanuele Dotto, Achim Krause, Thomas Nikolaus, and Irakli Patchkoria. Witt vectors with coefficients and characteristic polynomials over non-commutative rings. Compos. Math., 158(2):366–408, February 2022. doi:10.1112/S0010437X22007254.

$\bullet $ Fabian Hebestreit, Markus Land, and Thomas Nikolaus. On the homotopy type of L-spectra of the integers. J. Topol., 14(1):183–214, March 2021. doi:10.1112/topo.12180.

$\bullet $ Clark Barwick, Saul Glasman, Akhil Mathew, and Thomas Nikolaus. K-theory and polynomial functors. arXiv e-prints, February 2021. arXiv:2102.00936.

$\bullet $ Benjamin Antieau and Thomas Nikolaus. Cartier modules and cyclotomic spectra. Journal of the American Mathematical Society, 34(1):1–78, January 2021. doi:10.1090/jams/951.

$\bullet $ Baptiste Calmès, Emanuele Dotto, Yonatan Harpaz, Fabian Hebestreit, Markus Land, Kristian Moi, Denis Nardin, Thomas Nikolaus, and Wolfgang Steimle. Hermitian K-theory for stable ∞-categories III: Grothendieck-Witt groups of rings. arXiv e-prints, September 2020. arXiv:2009.07225.

$\bullet $ Baptiste Calmès, Emanuele Dotto, Yonatan Harpaz, Fabian Hebestreit, Markus Land, Kristian Moi, Denis Nardin, Thomas Nikolaus, and Wolfgang Steimle. Hermitian K-theory for stable ∞-categories I: Foundations. arXiv e-prints, September 2020. arXiv:2009.07223.

$\bullet $ Baptiste Calmès, Emanuele Dotto, Yonatan Harpaz, Fabian Hebestreit, Markus Land, Kristian Moi, Denis Nardin, Thomas Nikolaus, and Wolfgang Steimle. Hermitian K-theory for stable ∞-categories II: Cobordism categories and additivity. arXiv e-prints, September 2020. arXiv:2009.07224.

$\bullet $ Lars Hesselholt and Thomas Nikolaus. Algebraic $K$-theory of planar cuspidal curves. In $K$-theory in algebra, analysis and topology, volume 749 of Contemp. Math., pages 139–148. Amer. Math. Soc., [Providence], RI, March 2020. doi:10.1090/conm/749/15070.

$\bullet $ Benjamin Antieau, Akhil Mathew, Matthew Morrow, and Thomas Nikolaus. On the Beilinson fiber square. arXiv e-prints, March 2020. arXiv:2003.12541.

$\bullet $ Emanuele Dotto, Achim Krause, Thomas Nikolaus, and Irakli Patchkoria. Witt vectors with coefficients and characteristic polynomials over non-commutative rings. arXiv e-prints, February 2020. arXiv:2002.01538v1.

$\bullet $ Thomas Nikolaus and Konrad Waldorf. Higher geometry for non-geometric T-duals. Communications in Mathematical Physics, 374(1):317–366, February 2020. doi:10.1007/s00220-019-03496-3.

$\bullet $ Ulrich Bunke and Thomas Nikolaus. Twisted differential cohomology. Algebr. Geom. Topol., 19(4):1631–1710, August 2019. doi:10.2140/agt.2019.19.1631.

$\bullet $ Achim Krause and Thomas Nikolaus. Bökstedt periodicity and quotients of DVRs. arXiv e-prints, July 2019. arXiv:1907.03477.

$\bullet $ Tobias Barthel, Markus Hausmann, Niko Naumann, Thomas Nikolaus, Justin Noel, and Nathaniel Stapleton. The Balmer spectrum of the equivariant homotopy category of a finite abelian group. Inventiones Mathematicae, 216(1):215–240, April 2019. doi:10.1007/s00222-018-0846-5.