# apl. Prof. Dr. Lutz Hille, 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/hille/index.html |

Project membershipMathematics Münster | A: Arithmetic and GroupsC: Models and ApproximationsA1: Arithmetic, geometry and representations C1: Evolution and asymptotics |

Current Projects | • 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 • EXC 2044 - C1: Evolution and asymptotics In this unit, we will use generalisations of optimal transport metrics to develop gradient flow descriptions of (cross)-diffusion-reaction systems, rigorously analyse their pattern forming properties, and develop corresponding efficient numerical schemes. Related transport-type- and hyperbolic systems will be compared with respect to their pattern-forming behaviour, especially when mass is conserved. Bifurcations and the effects of noise perturbations will be explored.Moreover, we aim to understand defect structures, their stability and their interactions. Examples are the evolution of fractures in brittle materials and of vortices in fluids. Our analysis will explore the underlying geometry of defect dynamics such as gradient descents or Hamiltonian structures. Also, we will further develop continuum mechanics and asymptotic descriptions for multiple bodies which deform, divide, move, and dynamically attach to each other in order to better describe the bio-mechanics of growing and dividing soft tissues. Finally, we are interested in the asymptotic analysis of various random structures as the size or the dimension of the structure goes to infinity. More specifically, we shall consider random polytopes and random trees.For random polytopes we would like to compute the expected number of faces in all dimensions, the expected (intrinsic) volume, and absorption probabilities, as well as higher moments and limit distributions for these quantities. online |

E-Mail | lutz.hille@uni-muenster.de |

Phone | +49 251 83-33719 |

FAX | +49 251 83-33786 |

Room | 314 |

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

Address | apl. Prof. Dr. Lutz Hille Mathematisches Institut Fachbereich Mathematik und Informatik der Universität Münster Einsteinstrasse 62 48149 Münster Deutschland |

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