Sprechstunde nach Vereinbarung per E-Mail.
Private Homepagehttps://www.rzeidler.eu
Research InterestsScalar curvature
Spin geometry and Dirac operators
Index theory, K-theory and non-commutative geometry
Selected PublicationsCecchini 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 ProjectsComparison 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.

There are two main flavours of methods to probe the geometry of scalar curvature: One goes back to Lichnerowicz and uses various versions of index theory for the Dirac equation on spinors. The other is broadly based on minimal hypersurfaces and was initiated by Schoen and Yau. On both types of methods there has been tremendous progress over recent years sparked by novel quantitative comparison and rigidity questions due to Gromov and by on-going attempts to arrive at a deeper geometric understanding of lower scalar curvature bounds.

In this proposal we view established landmark results, such as the non-existence of positive scalar curvature on the torus, together with the more recent quantitative problems from a conceptually unified standpoint, where a comparison principle for scalar and mean curvature along maps between Riemannian manifolds plays the central role.

Guided by this point of view, we aim to develop fundamentally new tools to study scalar curvature that bridge long-standing gaps in between the existing techniques. This includes a far-reaching generalization of the Dirac operator approach expanding upon techniques pioneered by the PI, and novel applications of Bochner-type methods. We will also study analogous comparison problems on domains with singular boundary motivated by a first synthetic characterization of lower scalar curvature bounds in terms of polyhedral domains, and by the general quest for extending the study of scalar curvature beyond smooth manifolds. At the same time, we will treat subtle almost rigidity questions corresponding to the rigidity aspect of our comparison principle.

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
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
Project membership
Mathematics Münster


B: Spaces and Operators

B2: Topology
Current PublicationsCecchini, 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 ProjectsComparison 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.

There are two main flavours of methods to probe the geometry of scalar curvature: One goes back to Lichnerowicz and uses various versions of index theory for the Dirac equation on spinors. The other is broadly based on minimal hypersurfaces and was initiated by Schoen and Yau. On both types of methods there has been tremendous progress over recent years sparked by novel quantitative comparison and rigidity questions due to Gromov and by on-going attempts to arrive at a deeper geometric understanding of lower scalar curvature bounds.

In this proposal we view established landmark results, such as the non-existence of positive scalar curvature on the torus, together with the more recent quantitative problems from a conceptually unified standpoint, where a comparison principle for scalar and mean curvature along maps between Riemannian manifolds plays the central role.

Guided by this point of view, we aim to develop fundamentally new tools to study scalar curvature that bridge long-standing gaps in between the existing techniques. This includes a far-reaching generalization of the Dirac operator approach expanding upon techniques pioneered by the PI, and novel applications of Bochner-type methods. We will also study analogous comparison problems on domains with singular boundary motivated by a first synthetic characterization of lower scalar curvature bounds in terms of polyhedral domains, and by the general quest for extending the study of scalar curvature beyond smooth manifolds. At the same time, we will treat subtle almost rigidity questions corresponding to the rigidity aspect of our comparison principle.

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-Mailrudolf.zeidler@uni-muenster.de
Phone+49 251 83-33740
FAX+49 251 83-38370
Room517
Secretary   Sekretariat AG Topologie
Frau Claudia Rüdiger
Telefon +49 251 83-35159
Fax +49 251 83-38370
Zimmer 516
AddressProf. Dr. Rudolf Zeidler
Mathematisches Institut
Fachbereich Mathematik und Informatik der Universität Münster
Einsteinstrasse 62
48149 Münster
Deutschland
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