# PD Dr. Rudolf Zeidler, Mathematisches Institut

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

Investigator in Mathematics Münster

Sprechstunde nach Vereinbarung per E-Mail.

Private Homepage | https://www.rzeidler.eu |

Research Interests | Scalar curvature Spin geometry and Dirac operators Index theory, K-theory and non-commutative geometry |

Selected Publications | • Cecchini S, Zeidler R Scalar and mean curvature comparison via the Dirac operator. Geometry and Topology Vol. to appear, 2023 online• Zeidler R Band width estimates via the Dirac operator. Journal of Differential Geometry Vol. 122 (1), 2022 online• Xie Z, Yu G, Zeidler R On the range of the relative higher index and the higher rho-invariant for positive scalar curvature. Advances in Mathematics Vol. 390 (107897), 2021 online• Nitsche M, Schick T, Zeidler R Transfer maps in generalized group homology via submanifolds. Doc. Math. Vol. 26, 2021, pp 947-979 online• Engel A, Wulff C, Zeidler R Slant products on the Higson-Roe exact sequence. Ann. Inst. Fourier Vol. 71 (3), 2021, pp 913-1021 online• Zeidler R Width, Largeness and Index Theory. SIGMA Vol. 16, 2020, pp 15 pages online• Bárcenas N, Zeidler R Positive scalar curvature and low-degree group homology. Annals of K-theory Vol. 3 (3), 2018, pp 565-579 online• Zeidler R An index obstruction to positive scalar curvature on fiber bundles over aspherical manifolds. Algebraic and Geometric Topology Vol. 17 (5), 2017, pp 3081-3094 online• Zeidler R Positive scalar curvature and product formulas for secondary index invariants. Journal of Topology Vol. 9 (3), 2016, pp 687-724 online• Cecchini S, Zeidler R The positive mass theorem and distance estimates in the spin setting. Preprint Vol. Preprint online |

Selected Projects | • SPP 2026: Geometry at Infinity - Subproject: Duality and the coarse assembly map The coarse co-assembly map was introduced by Emerson and Meyer as a dual to the coarse assembly map. These two maps are mutually adjoint with respect to canonical pairings. The main objective of our project is to further develop this coarse duality theory by studying analogues of multiplicative structures known from algebraic topology such as cup and cap products as well as external and slant products. Many of these multiplicative structures have direct interpretations in terms of Dirac operators and vector bundles leading to new applications in index theory and geometric topology. In addition, we will develop coarse versions of well-known results from algebraic topology involving multiplicative structures.Most notably, we want to investigate a coarse version of Poincaré duality.Furthermore, the question of whether coarse assembly and co-assembly are isomorphisms will also be examined in certain cases. We have a special focus on spaces which admit nice coarse compactifications constructed from coarse geometric versions of contractions or, more generally, deformation retractions. Such coarse deformation retractions are also key to the construction of secondary cup and cap products, establishing a close link between the two main aspects of our project. 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 |

Project membershipMathematics Münster | B: Spaces and OperatorsB2: Topology |

Current Publications | • Cecchini S, Zeidler R Scalar and mean curvature comparison via the Dirac operator. Geometry and Topology Vol. to appear, 2023 online• Zeidler R Band width estimates via the Dirac operator. Journal of Differential Geometry Vol. 122 (1), 2022 online• Nitsche M, Schick T, Zeidler R Transfer maps in generalized group homology via submanifolds. Doc. Math. Vol. 26, 2021, pp 947-979 online• Xie Z, Yu G, Zeidler R On the range of the relative higher index and the higher rho-invariant for positive scalar curvature. Advances in Mathematics Vol. 390 (107897), 2021 online• Engel A, Wulff C, Zeidler R Slant products on the Higson-Roe exact sequence. Ann. Inst. Fourier Vol. 71 (3), 2021, pp 913-1021 online• Zeidler R Width, Largeness and Index Theory. SIGMA Vol. 16, 2020, pp 15 pages online• Bárcenas N, Zeidler R Positive scalar curvature and low-degree group homology. Annals of K-theory Vol. 3 (3), 2018, pp 565-579 online |

Current Projects | • SPP 2026: Geometry at Infinity - Subproject: Duality and the coarse assembly map The coarse co-assembly map was introduced by Emerson and Meyer as a dual to the coarse assembly map. These two maps are mutually adjoint with respect to canonical pairings. The main objective of our project is to further develop this coarse duality theory by studying analogues of multiplicative structures known from algebraic topology such as cup and cap products as well as external and slant products. Many of these multiplicative structures have direct interpretations in terms of Dirac operators and vector bundles leading to new applications in index theory and geometric topology. In addition, we will develop coarse versions of well-known results from algebraic topology involving multiplicative structures.Most notably, we want to investigate a coarse version of Poincaré duality.Furthermore, the question of whether coarse assembly and co-assembly are isomorphisms will also be examined in certain cases. We have a special focus on spaces which admit nice coarse compactifications constructed from coarse geometric versions of contractions or, more generally, deformation retractions. Such coarse deformation retractions are also key to the construction of secondary cup and cap products, establishing a close link between the two main aspects of our project. 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 | rudolf.zeidler@uni-muenster.de |

Phone | +49 251 83-33740 |

FAX | +49 251 83-38370 |

Room | 517 |

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

Address | PD Dr. Rudolf Zeidler Mathematisches Institut Fachbereich Mathematik und Informatik der Universität Münster Einsteinstrasse 62 48149 Münster Deutschland |

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