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Mathematics Münster

B: Spaces and Operators

B2: Topology
Current PublicationsBartels A, Bestvina M The Farrell-Jones conjecture for mapping class groups. Inventiones Mathematicae Vol. 215 (2), 2019, pp 651-712 online
Bartels A, Douglas CL, Henriques A Fusion of defects. Memoirs of the American Mathematical Society Vol. 258 (1237), 2019, pp vii+100 online
Bartels A, Douglas CL, Henriques A Conformal nets IV: The 3-category. Algebraic and Geometric Topology Vol. 18 (2), 2018, pp 897-956 online
Current ProjectsCRC 1442: Geometry: Deformation and Rigidity - C03: K-theory of group algebras The Farrell–Jones conjecture gives a homological formula for the K-theory of group rings. On the one hand this conjecture has concrete applications, e.g. to the rigidity of aspherical manifolds. On the other hand its underlying principle is very general, and applies for example also to Hecke algebras of p-adic Lie groups. We will extend the scope of positive results for this conjecture, develop new tools to study the conjecture and extend the framework where is can be formulated and applied. online
CRC 1442: Geometry: Deformation and Rigidity - Z01: Central Task of the Collaborative Research Centre online
CRC 1442: Geometry: Deformation and Rigidity From its historic roots, geometry has evolved into a cornerstone in modern mathematics, both as a tool and as a subject in its own right. On the one hand many of the most important open questions in mathematics are of geometric origin, asking for example to what extent an object is determined by geometric properties. On the other hand, abstract mathematical problems can often be solved by associating them to more geometric objects that can then be investigated using geometric tools. A geometric point of view on an abstract mathematical problem quite often opens a path to its solution.

Deformations and rigidity are two antagonistic geometric concepts which can be applied in many abstract situations making transfer of methods particularly fruitful. Deformations of mathematical objects can be viewed as continuous families of such objects, like for instance evolutions of a shape or a system with time. The collection of all possible deformations of a mathematical object can often be considered as a deformation space (or moduli space), thus becoming a geometric object in its own right. The geometric properties of this space in turn shed light on the deeper structure of the given mathematical objects. We think of properties or of quantities associated with mathematical objects as rigid if they are preserved under all (reasonable) deformations. A rigidity phenomenon refers to a situation where essentially no deformations are possible. Rigidity then implies that objects which are approximately the same must in fact be equal, making such results important for classifications.

The overall objective of our research programme can be summarised as follows:

Develop geometry as a subject and as a powerful tool in theoretical mathematics focusing on the dichotomy of deformations versus rigidity. Use this unifying perspective to transfer deep methods and insights between different mathematical subjects to obtain scientific breakthroughs, for example concerning the Langlands programme, positive curvature manifolds, K-theory, group theory, and C*-algebras. 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-Mailbartelsa at math dot uni-muenster dot de
Phone+49 251 83-33745
FAX+49 251 83-32774
Secretary   Sekretariat AG Topologie
Frau Claudia Rüdiger
Telefon +49 251 83-35159
Fax +49 251 83-38370
Zimmer 516
Sekretariat SFB 1442

Telefon +49 251 83-
Fax +49 251 83-
Sekretariat des Sonderforschungsbereichs 1442
AddressProf. Dr. Arthur Bartels
Mathematisches Institut
Fachbereich Mathematik und Informatik der Universität Münster
Einsteinstrasse 62
48149 Münster
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