# Prof. Dr. Thomas Nikolaus, Mathematisches Institut

Member of CRC 1442 Geometry: Deformations and RigidityInvestigator in Mathematics Münster

Investigator in Mathematics Münster

Private Homepage | https://wwwmath.uni-muenster.de/u/nikolaus |

Research Interests | Topology Homotopy Theory Arithmetic Higher Categories |

Project membershipMathematics Münster | A: Arithmetic and GroupsA1: Arithmetic, geometry and representations |

Current Talks | • Frobenius homomorphisms in higher algebra. International Congress of Mathematics 2022, Virtual event Slides Link to event |

Current Publications | • Krause, 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 Projects | • CRC 1442: Geometry: Deformation and Rigidity - A04: New cohomology theories for arithmetic schemes The goal of this project is to study cohomology theories for schemes in order to attack important open problems in arithmetic. Among these theories are topological periodic homology (TP), topological cyclic homology (TC), rational de Rham–Witt cohomology, prismatic cohomology, K-theory, L-Theory and leafwise cohomology of associated dynamical systems. We will prove structural results about those theories as well as make further calculations of specific cases. • CRC 1442: Geometry: Deformation and Rigidity - C02: Homological algebra for stable ∞-categories The general goal of the project is to study the homological algebra of stable infinity-categories and Poincaré infinity-categories. This is done through the theory of non-commutative motives and Efimov K-Theory. Concrete goals are to give a new approach to controlled algebra (thereby attacking open problems and conjectures in geometric topology) and obtain new structural results about the category of motives. The latter thus yields new results about K-Theory and TC, e.g. we try to resolve the long-standing open question about a universal property for TC. • 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 |

E-Mail | nikolaus@uni-muenster.de |

Phone | +49 251 83-33744 |

FAX | +49 251 83-38370 |

Room | 515 |

Secretary | Sekretariat AG Topologie Frau Claudia Rüdiger Telefon +49 251 83-35159 Fax +49 251 83-38370 Zimmer 516 |

Address | Prof. Dr. Thomas Nikolaus Mathematisches Institut Fachbereich Mathematik und Informatik der Universität Münster Einsteinstrasse 62 48149 Münster Deutschland |

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