# Prof. Dr. Eugen Hellmann, Mathematisches Institut

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

Investigator in Mathematics Münster

Private Homepage | https://www.uni-muenster.de/Arithm/hellmann/index.html |

Research Interests | Arithmetische Algebraische Geometrie Langlands Programm p-adische Hodge Theorie und p-adische Galoisdarstellungen p-adische automorphe Formen und p-adisches Langlands Programm |

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

Current Publications | • Hellmann, Eugen On the derived category of the Iwahori-Hecke algebra. Compositio Mathematica Vol. 159 (5), 2023 online• Hellmann, Eugen Density of automorphicpoints in deformation rings of polarized global Galois representations. Duke Mathematical Journal Vol. 171 (13), 2022 online• Hellmann, Eugen
Hartl, Urs The universal family of semi-stable p-adic Galois representations. Algebra and Number Theory Vol. 14 (5), 2020 online• Hellmann, Eugen A local model for the triangulate variety and applications. Publications Mathématiques de L'IHÉS Vol. 130, 2019 online |

Current Projects | • CRC 1442: Geometry: Deformation and Rigidity From its historic roots, geometry has evolved into a cornerstone in modern mathematics, both as a tool and as a subject in its own right. On the one hand many of the most important open questions in mathematics are of geometric origin, asking for example to what extent an object is determined by geometric properties. On the other hand, abstract mathematical problems can often be solved by associating them to more geometric objects that can then be investigated using geometric tools. A geometric point of view on an abstract mathematical problem quite often opens a path to its solution.Deformations and rigidity are two antagonistic geometric concepts which can be applied in many abstract situations making transfer of methods particularly fruitful. Deformations of mathematical objects can be viewed as continuous families of such objects, like for instance evolutions of a shape or a system with time. The collection of all possible deformations of a mathematical object can often be considered as a deformation space (or moduli space), thus becoming a geometric object in its own right. The geometric properties of this space in turn shed light on the deeper structure of the given mathematical objects. We think of properties or of quantities associated with mathematical objects as rigid if they are preserved under all (reasonable) deformations.A rigidity phenomenon refers to a situation where essentially no deformations are possible.Rigidity then implies that objects which are approximately the same must in fact be equal, making such results important for classifications.The overall objective of our research programme can be summarised as follows:Develop geometry as a subject and as a powerful tool in theoretical mathematics focusing on the dichotomy of deformations versus rigidity. Use this unifying perspective to transfer deep methods and insights between different mathematical subjects to obtain scientific breakthroughs, for example concerning the Langlands programme, positive curvature manifolds, K-theory, group theory, and C*-algebras. • CRC 1442: Geometry: Deformation and Rigidity - A01: Automorphic forms and the p-adic Langlands programme The past years have seen tremendous progress in the development of a categorical approach to the arithmetic of the Langlands programme. In the context of the p-adic Langlands programme the main features of this approach are the study of derived categories of p-adic representations of p-adic Lie groups, the study of (coherent) sheaves on moduli of Galois representations associated to such representations, and the development of a more geometric approach to such representations. The project addresses all these three aspects of the programme. • CRC 1442: Geometry: Deformation and Rigidity - A02: Moduli spaces of p-adic Galois representations Representations of the absolute Galois group of a p-adic local field with p-adic coefficients are studied most fruitfully in terms of semi-linear algebra objects called (phi,Gamma)-modules. In part of the project we will advance the study of (phi,Gamma)-modules. In another part we use (phi,Gamma)-modules to construct and study moduli spaces of Galois representation that occur in the context of the p-adic Langlands programme. • CRC 1442: Geometry: Deformation and Rigidity - Z01: Central Task of the Collaborative Research Centre 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 |

E-Mail | e.hellmann@uni-muenster.de |

Phone | +49 251 83-33723 |

FAX | +49 251 83-33786 |

Room | 318 |

Secretary | Sekretariat Harenbrock/Reckermann Frau Ina Reckermann Telefon +49 251 83-33700 Fax +49 251 83-33786 Zimmer 316 |

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

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