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Research InterestsTopology
Homotopy Theory
Higher Categories
Project membership
Mathematics Münster

A: Arithmetic and Groups

A1: Arithmetic, geometry and representations
Current TalksFrobenius homomorphisms in higher algebra. International Congress of Mathematics 2022, Virtual event Slides Link to event
Current PublicationsKrause, Achim; McCandless, Jonas; Nikolaus, Thomas Polygonic spectra and TR with coefficients. , 2023 online
Dotto E., Krause A., Nikolaus T., Patchkoria I. Witt vectors with coefficients and characteristic polynomials over non-commutative rings. Compositio Mathematica Vol. 158 (2), 2022, pp 366-408 online
Antieau, Benjamin; Krause, Achim; Nikolaus, Thomas On the K-theory of Z/pn. , 2022 online
Land, Markus; Nikolaus, Thomas; Schlichting, Marco L-theory of C∗-algebras. , 2022 online
Antieau B., Nikolaus T. Cartier modules and cyclotomic spectra. Journal of the American Mathematical Society Vol. 34 (1), 2021, pp 1-78 online
Hebestreit F., Land M., Nikolaus T. On the homotopy type of L-spectra of the integers. Journal of Topology Vol. 14 (1), 2021, pp 183-214 online
Barwick, Clark; Glasman, Saul; Mathew, Akhil; Nikolaus, Thomas K-theory and polynomial functors. , 2021 online
Hesselholt L., Nikolaus T. Algebraic k-theory of planar cuspidal curves. K-Theory in Algebra, Analysis and TopologyContemporary MathematicsContemporary Mathematics Vol. 749, 2020, pp 139-148 online
Nikolaus T., Waldorf K. Higher Geometry for Non-geometric T-Duals. Communications in Mathematical Physics Vol. 374 (1), 2020, pp 317-366 online
Current ProjectsCRC 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
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
EXC 2044 - A1: Arithmetic, geometry and representations The Langlands programme relates representations of (the adele valued points of) reductive groups G over Q - so-called automorphic representations - with certain representations of the absolute Galois group of Q. This programme includes the study of these objects over general global fields (finite extension of Q or Fp (t)) and local fields as well. In its local form the classical programme onlyconsidered l-adic Galois representations of p-adic fields for unequal primes l neq p. In order to allow for a p-adic variation of the objects, it is absolutely crucial to extend it to the case l = p. In the global situation, the automorphic representations in question can often be realised in (or studied via) the cohomology of a tower of Shimura varieties (or related moduli spaces) attached to the group G. We will focus on the following directions within this programme: The p-adic and mod p Langlands programme asks for an extension of such a correspondence involving certain continuous representations with p-adic respectively mod p coefficients. Broadening the perspective to p-adic automorphic forms should, for example, enable us to capture all Galois representations, not just those having a particular Hodge theoretic behaviour at primes dividing p. This extended programme requires the introduction of derived categories. We will study differential graded Hecke algebras and their derived categories on the reductive group side. On the Galois side, we hope to use derived versions of the moduli spaces of p-adic Galois representations introduced by Emerton and Gee. The geometric Langlands programme is a categorification of the Langlands programme. We plan to unify the different approaches using motivic methods. In another direction, we study the geometry and arithmetic of moduli stacks of global G-shtukas over function fields. Their cohomology has been the crucial tool to establish large parts of the local and global Langlands programme over function fields. Variants of G-shtukas are also used to construct and investigate families of p-adic Galois representations.

Cohomology theories are a universal tool pervading large parts of algebraic and arithmetic geometry. We will develop and study cohomology theories, especially in mixed characteristic, that generalise and unify étale cohomology, crystalline cohomology and de Rham cohomology as well as Hochschild cohomology in the non-commutative setting. Developing (topological) cyclic homology in new contexts is an important aim. A main goal is to construct a cohomology theory that can serve the same purposes for arithmetic schemes as the l-adic or crystalline cohomology with their Frobenius actions for varieties over finite fields. Ideas from algebraic geometry, algebraic topology, operator algebras and analysis blend in these investigations. online
Cluster of Excellence 2044 - Mathematics Münster: Dynamics – Geometry – Structure We will tackle fundamentally important mathematical problems, viewing mathematics as an organic whole with countless interactions. Our research is unified by three major approaches: focusing on the underlying structure of a given problem, taking the geometric viewpoint and studying the relevant dynamics of group or semigroup actions. The theories which we will build will not only solve the problems under consideration but also many others of a similar nature; these theories will also raise exciting new questions. online
Phone+49 251 83-33744
FAX+49 251 83-38370
Secretary   Sekretariat AG Topologie
Frau Claudia Rüdiger
Telefon +49 251 83-35159
Fax +49 251 83-38370
Zimmer 516
AddressProf. Dr. Thomas Nikolaus
Mathematisches Institut
Fachbereich Mathematik und Informatik der Universität Münster
Einsteinstrasse 62
48149 Münster
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