# Prof. Dr. Rudolf Zeidler, Mathematisches Institut

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

Field of expertise: Topology

Investigator in Mathematics Münster

Field of expertise: Topology

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 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, Alexander; Wulff, Christopher; Zeidler, Rudolf Slant products on the Higson-Roe exact sequence. Annales de l’Institut 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. Transactions of the American Mathematical Society online |

Selected Projects | • Comparison and rigidity for scalar curvature Questions involving the scalar curvature bridge many areas inside mathematics including geometric analysis, differential geometry and algebraic topology, and they are naturally related to the mathematical description of general relativity. • 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 • SPP 2026 - 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 |

Topics inMathematics Münster | T1: K-Groups and cohomologyT2: Moduli spaces in arithmetic and geometryT5: Curvature, shape, and global analysis |

Current Publications | • Cecchini, Simone; Lesourd, Martin; Zeidler, Rudolf Positive Mass Theorems for Spin Initial Data Sets With Arbitrary Ends and Dominant Energy Shields. International Mathematics Research Notices Vol. 2024 (9), 2024 online• Cecchini, Simone; Räde, Daniel; Zeidler, Rudolf Nonnegative scalar curvature on manifolds with at least two ends. Journal of Topology Vol. 16 (3), 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, Alexander; Wulff, Christopher; Zeidler, Rudolf Slant products on the Higson-Roe exact sequence. Annales de l’Institut 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 | • CRC 1442 - B03: Moduli spaces of metrics of positive curvature We will develop a family version of coarse index theory which encompasses all existing index invariants for the understanding of spaces of positive scalar curvature (psc) metrics—the higher family index and index difference—as well as new ones such as family rho-invariants. This will enable the detection of new non-trivial elements in homotopy groups of certain moduli spaces of psc metrics. We will also further study the concordance space of psc metrics together with appropriate index maps. • CRC 1442 - B05: Scalar curvature between Kähler and spin This project aims to connect recent developments in Kähler geometry and spin geometry related to lower scalar curvature bounds and the Positive Mass Theorem. We would like to sharpen the Cecchini-Zeidler bandwidth inequality in the case of Kähler metrics and to find new proofs and extensions of the Positive Mass Theorem. One setting of interest is the case of spin^c manifolds equipped with almost-Kähler metrics, particularly in real dimension 4. • Comparison and rigidity for scalar curvature Questions involving the scalar curvature bridge many areas inside mathematics including geometric analysis, differential geometry and algebraic topology, and they are naturally related to the mathematical description of general relativity. • 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 | Prof. Dr. Rudolf Zeidler Mathematisches Institut Fachbereich Mathematik und Informatik der Universität Münster Einsteinstrasse 62 48149 Münster Deutschland |

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