# Prof. Dr. Giles Gardam, Mathematisches Institut

Member of CRC 1442 Geometry: Deformations and RigidityMember of Mathematics Münster

Investigator in Mathematics Münster

Member of Mathematics Münster

Investigator in Mathematics Münster

Private Homepage | https://www.gilesgardam.com |

Research Interests | Geometric group theory Group rings |

Project membershipMathematics Münster | A: Arithmetic and GroupsB: Spaces and OperatorsA2: Groups, model theory and sets B2: Topology |

Current Publications | • Benjamin, Beeker; Cordes, Matthew; Gardam, Giles; Gupta, Radhika; Stark, Emily Cannon-Thurston maps for CAT(0) groups with isolated flats. Mathematische Annalen Vol. 384 (1-2), 2022 online• Gardam Giles A counterexample to the unit conjecture for group rings. Annals of Mathematics Vol. 194 (2021) (3), 2021, pp 967-979 online |

Current Projects | • EXC 2044 - A2: Groups, model theory and sets Model theory and, more generally, mathematical logic as a whole has seen striking applicationsto arithmetic geometry, topological dynamics and group theory. Such applications as well asfundamental questions in geometric group theory, on the foundations of model theory and in settheory are at the focus of our work. The research in our group reaches from group theoretic questions to the model theory of groups andvalued fields as well as set theory. The study of automorphism groups of first order structures astopological groups, for examples, uses tools from descriptive set theory leading to pure set theoretic questions. online• EXC 2044 - B2: Topology We will analyse geometric structures from a topological point of view. In particular, we will study manifolds, their diffeomorphisms and embeddings, and positive scalar curvature metrics on them. Via surgery theory and index theory many of the resulting questions are related to topological K-theory of group C*-algebras and to algebraic K-theory and L-theory of group rings.Index theory provides a map from the space of positive scalar curvature metrics to the K-theory of the reduced C*-algebra of the fundamental group. We will develop new tools such as parameterised coarse index theory and combine them with cobordism categories and parameterised surgery theory to study this map. In particular, we aim at rationally realising all K-theory classes by families of positive scalar curvature metrics. Important topics and tools are the isomorphism conjectures of Farrell-Jones and Baum-Connes about the structure of the K-groups that appear. We will extend the scope of these conjectures as well as the available techniques, for example from geometric group theory and controlled topology. We are interested in the K-theory of Hecke algebras of reductive p-adic Lie groups and in the topological K-theory of rapid decay completions of complex group rings. Via dimension conditions and index theory techniques we will exploit connections to operator algebras and coarse geometry. Via assembly maps in algebraic K- and L-theory we will develop index-theoretic tools to understand tangential structures of manifolds. We will use these tools to analyse the smooth structure space of manifolds and study diffeomorphism groups. Configuration categories will be used to study spaces of embeddings of manifolds. online |

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

Phone | +49 251 83-33757 |

FAX | +49 251 83-38370 |

Room | 513 |

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

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

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