TRR 326:  Geometry and Arithmetic of Uniformized Structures (GAUS)

• Subject Area: Mathematics
• Term: since 2021
• Website: Homepage
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 444845124

The concept of uniformization is ubiquitous in mathematics. It serves as a tool to replace a com- plicated geometric object by a simpler one without altering the local structure. The original mathe¬matical complexity is now encoded in a suitable symmetry group. This translation into another language opens up new perspectives for the study of the original mathematical objects. A very active and successful area of research uses this rich framework to study the geometry and arithmetic of algebraic varieties. Important features in the concept of uniformized spaces include the following: automorphic forms arise as functions respecting these symmetry groups. Galois representations, which are fundamental in all branches of modern number theory, encode arithmetic symmetries. The tower of all covering spaces between the original object and its uniformization reflects the ‘paths’ of the space under consideration and ultimately leads to topological and cohomological invariants that are often of a non-abelian nature. Frobenius symmetry enhances geometry in the parallel universe of positive characteristic in an unexpected manner. Celebrated new results by Fields medallist Peter Scholze provide a bridge from the classical setting to the realm of positive characteristic, thus allowing the application of Frobenius symmetry even to classical questions in the framework of uniformization. The CRC/TRR will play a leading role in future developments regarding all branches of arithmetic and geometry in which uniformization manifests as a key structure. The aim is twofold: We want to further extend and develop the techniques of uniformization in various settings, and we also want to apply them to central geometric and arithmetic questions, notably regarding the global geometry of moduli spaces and Shimura varieties, the arithmetic complexity of symmetry groups and the interplay between geometry and Galois representations. The core areas of the CRC/TRR are: (A) Moduli spaces and automorphic forms,(B) Galois representations and étale invariants,(C) Cohomological structures and degeneration in positive characteristic. The focused research activities in Darmstadt, Frankfurt, and Heidelberg will lead to significant progress in topical areas of Arithmetic Algebraic Geometry, including but not limited to the Kudla programme, Teichmüller geometry, aspects of the local Langlands programme, anabelian geometry, and Iwasawa theory. Within the LOEWE research unit at Darmstadt/Frankfurt and the DFG research unit at Darm- stadt/Heidelberg we have formed a group of internationally renowned and well-connected experts. The CRC/TRR strengthens these efforts by joining the two groups and by expanding the scope of their research on the geometry and arithmetic of uniformized structures.

• DFG Programme: CRC/Transregios

Current projects

• A01 - Teichmüller geometry of the moduli space (Project Head Möller, Martin )
• A02 - Non-archimedean and tropical geometry of moduli spaces (Project Heads Möller, Martin ;  Ulirsch, Ph.D., Martin ;  Werner, Annette )
• A03 - Non-archimedean skeletons and Newton–Okounkov bodies (Project Heads Küronya, Alex ;  Ulirsch, Ph.D., Martin )
• A04 - Green currents on Shimura varieties (Project Head Bruinier, Jan Hendrik )
• A05 - Expansion and rationality of theta integrals (Project Head Li, Ph.D., Yingkun )
• A06 - Automorphic forms and vertex operator algebras (Project Head Scheithauer, Nils R. )
• A07 - Cusp forms on Drinfeld period domains (Project Head Böckle, Gebhard )
• A08 - Geodesic cycles and modular forms (Project Heads Bruinier, Jan Hendrik ;  Möller, Martin )
• A09 - Effective global generation for uniformized varieties (Project Heads Küronya, Alex ;  Stix, Jakob )
• B01 - Higher dimensional anabelian geometry (Project Heads Schmidt, Alexander ;  Stix, Jakob )
• B02 - Galois representations in anabelian geometry (Project Head Stix, Jakob )
• B03 - Motivic local systems of Calabi–Yau-type (Project Head van Straten, Duco )
• B04 - Images of Galois representations and deformations (Project Head Böckle, Gebhard )
• B05 - Iwasawa cohomology of Galois representations (Project Head Venjakob, Otmar )
• B06 - L-packets of p-adic automorphic forms (Project Head Ludwig, Judith )
• B07 - Motives and the Langlands programme (Project Head Richarz, Timo )
• C01 - Tame cohomology of schemes and adic spaces (Project Heads Hübner, Katharina ;  Schmidt, Alexander )
• C02 - Duality with Frobenius and Fp-étale ccohomology (Project Heads Blickle, Ph.D., Manuel ;  Böckle, Gebhard )
• C03 - Derived and prismatic F-Zips (Project Heads Blickle, Ph.D., Manuel ;  Wedhorn, Torsten )
• C04 - Motives for shtukas and Shimura varieties (Project Heads Richarz, Timo ;  Viehmann, Eva ;  Wedhorn, Torsten )
• C05 - Strata and tautological classes for compactifications of Shimura varieties (Project Head Wedhorn, Torsten )
• C06 - P-adic degeneration of vector bundles (Project Head Werner, Annette )
• MGK - Integrated Research Training Group (Project Head Wedhorn, Torsten )
• Z - Central Tasks of the Collaborative Research Centre (Project Head Stix, Jakob )

• Applicant Institution: Goethe-Universität Frankfurt am Main
• Co-Applicant Institution: Ruprecht-Karls-Universität Heidelberg; Technische Universität Darmstadt
• Participating University: Johannes Gutenberg-Universität Mainz; Technische Universität München (TUM), until 1/2022
• Spokesperson: Professor Dr. Jakob Stix