SFB 1442:  Geometry: Deformations and Rigidity

• Subject Area: Mathematics
• Term: since 2020
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 427320536

From its historic roots, geometry has evolved into a central subject in modern mathematics, both as a tool and as a subject in its own right. A vast number of important questions in mathematics have a genuinely geometric origin. On the other hand, abstract mathematical problems can often be fruitfully investigated by formulating them geometrically. The research programme of the CRC proposes to approach a variety of mathematical problems geometrically from two seemingly antagonistic but complementary poles: Deformations and Rigidity. Deformations of mathematical objects can be viewed as continuous families of these. Deformations exist not only for geometric objects, but for many other mathematical objects as well. Conversely, a rigidity phenomenon refers to a situation where essentially no deformations are possible: Properties or quantities associated with mathematical objects are rigid if they are preserved under all reasonable deformations. Rigidity then implies that objects which are approximately the same must in fact be equal, making such results important for classifications.The dichotomy of deformations and rigidity appears in the study of various geometric contexts in mathematics, notably in the Langlands programme, positive curvature manifolds, partial differential equations, K-theory, group theory, and C*-algebras. These research directions are the cornerstones of our proposal, all subject to rapid international developments. While deformations and rigidity in these contexts have usually been considered rather independently, the approach of our CRC is to use them as a strong guiding idea. In the past funding period this resulted in strong theorems and various interactions between the concrete projects connecting researchers from different backgrounds. Building on the new perspectives unravelled by the unified approach using deformations and rigidity, the CRC aims to contribute fundamental results and insights in the proposed second funding period. Overall the objective of our research programme can be summarised as follows: We aim to use the unifying perspective of deformations and rigidity to transfer deep methods and insights between different mathematical subjects to obtain scientific breakthroughs, for example concerning the Langlands programme, positive curvature manifolds, partial differential equations, K-theory, group theory, and C*-algebras.

• DFG Programme: Collaborative Research Centres
• International Connection: United Kingdom

Current projects

• A01 - Automorphic forms and the p-adic Langlands programme (Project Heads Hellmann, Eugen ;  Lourenco, Joao ;  Schneider, Peter )
• A02 - Moduli spaces of p-adic Galois representations (Project Heads Hartl, Urs ;  Hellmann, Eugen ;  Schneider, Peter )
• A04 - New cohomology theories for arithmetic schemes (Project Heads Deninger, Christopher ;  Nikolaus, Thomas )
• A05 - Moduli spaces of local shtukas in mixed characteristic (Project Heads Lourenco, Joao ;  Viehmann, Eva ;  Zhao, Yifei )
• B01 - Curvature and Symmetry (Project Heads Wiemeler, Michael ;  Wilking, Burkhard )
• B02 - Geometric evolution equations (Project Heads Böhm, Christoph ;  Wilking, Burkhard )
• B03 - Moduli spaces of metrics of positive curvature (Project Heads Ebert, Johannes ;  Zeidler, Rudolf )
• B04 - Geometric PDEs and Symmetry (Project Heads Böhm, Christoph ;  Siffert, Anna )
• B05 - Scalar curvature between Kähler and spin (Project Heads Hein, Hans-Joachim ;  Santoro, Bianca ;  Zeidler, Rudolf )
• B06 - Einstein 4-manifolds with two commuting Killing vectors (Project Heads Hein, Hans-Joachim ;  Holzegel, Gustav )
• B07 - Analytic and topological torsion (Project Heads Deninger, Christopher ;  Ludwig, Ursula )
• C02 - Homological algebra for stable ∞-categories (Project Head Nikolaus, Thomas )
• C03 - K-theory of group algebras (Project Head Bartels, Arthur )
• C04 - Group theoretic aspects of negative curvature (Project Head Tent, Katrin )
• D01 - Cartan sub-C*-algebras: an amenable perspective (Project Heads Geffen, Ph.D., Shirly ;  Winter, Wilhelm )
• D03 - Integrability (Project Heads Schürmann, Jörg ;  Wulkenhaar, Raimar ;  Zhao, Yifei )
• D04 - Entropy, orbit equivalence, and Bernoulli rigidity (Project Head Kerr, David )
• D05 - C*-algebras, groups, and dynamics: beyond amenability (Project Heads Geffen, Ph.D., Shirly ;  Kerr, David ;  Winter, Wilhelm )
• Z01 - Central Task of the Collaborative Research Centre (Project Heads Bartels, Arthur ;  Hellmann, Eugen )

Completed projects

• A03 - Special cycles on moduli spaces of G-shtukas (Project Head Hartl, Urs )
• C01 - Automorphisms and embeddings of manifolds (Project Heads Ebert, Johannes ;  Weiss, Michael )
• C05 - Rigidity of group topologies and universal minimal flows (Project Head Kwiatkowska, Ph.D., Aleksandra )
• D02 - Exotic crossed products and the Baum–Connes conjecture (Project Head Echterhoff, Siegfried )

• Applicant Institution: Universität Münster
• Spokesperson: Professor Dr. Eugen Hellmann