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

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

The concept of uniformization is ubiquitous in mathematics. It serves as a tool to replace a complicated 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 object. 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. Motives and homotopy theory provide new tools to approximate algebraic and topological invariants. Frobenius symmetry enhances geometry in the parallel universe of positive characteristic in an unexpected manner. Interpolating between positive characteristic and p-adic analytic settings, the celebrated new theory of prisms allows to apply Frobenius symmetry even to classical questions in the framework of uniformization. The CRC/TRR is playing 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, enumerative geometry, 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. In the first funding period the focused research activities in Darmstadt, Frankfurt, and Heidelberg have led to significant progress in topical areas of Arithmetic Algebraic Geometry, including but not limited to the Kudla programme, vertex operator algebras, local models of Shimura varieties, the local Langlands programme, anabelian geometry, Iwasawa theory, and p-adic non-abelian Hodge theory. Building on a tradition of collaboration in a LOEWE Research Unit at Darmstadt/Frankfurt and a DFG Research Unit at Darmstadt/Heidelberg we have complemented a group of internationally renowned and well-connected experts by strategic hirings. In the second funding period this will enable us to further expand the scope of our research on geometry and arithmetic of uniformized structures.

• DFG Programme: CRC/Transregios

Current projects

• A01 - Teichmüller geometry of the moduli space (Project Head Möller, Martin )
• 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 )
• A06 - Automorphic forms and vertex operator algebras (Project Head Scheithauer, Nils R. )
• A07 - Rigid meromorphic cocycles on Drinfeld period domains (Project Heads Böckle, Gebhard ;  Ludwig, Judith )
• 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 )
• A10 - Gromov–Witten theory and orthogonal modular forms (Project Heads Bruinier, Jan Hendrik ;  Oberdieck, Georg )
• A11 - Tropical correspondences for A1-enumerative geometry (Project Heads Möller, Martin ;  Pauli, Sabrina )
• A12 - Algebraic surgery and enumerative geometry (Project Heads Bachmann, Tom ;  Pauli, Sabrina )
• B01 - Higher dimensional anabelian geometry (Project Heads Schmidt, Alexander ;  Stix, Jakob )
• B02 - Galois representations in anabelian geometry (Project Head Stix, Jakob )
• 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 - Algebraic cobordism in geometric Langlands (Project Heads Bachmann, Tom ;  Eberhardt, Jens Niklas ;  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 - Truncations of shtukas and Shimura varieties (Project Heads Richarz, Timo ;  Viehmann, Eva ;  Wedhorn, Torsten )
• C06 - p-adic non-abelian Hodge theory (Project Heads Heuer, Ben ;  Werner, Annette )
• C07 - Motivic homotopy in p-adic cohomology (Project Head Merici, Alberto )
• C08 - K-theory, regularity, and homotopy invariance (Project Heads Hübner, Katharina ;  Tamme, Georg )
• MGK - Integrated Research Training Group (Project Head Wedhorn, Torsten )
• Z - Central Tasks of the Collaborative Research Centre (Project Head Stix, Jakob )

Completed projects

• A02 - Non-archimedean and tropical geometry of moduli spaces (Project Heads Möller, Martin ;  Ulirsch, Ph.D., Martin ;  Werner, Annette )
• A05 - Expansion and rationality of theta integrals (Project Head Li, Ph.D., Yingkun )
• B03 - Motivic local systems of Calabi–Yau-type (Project Head van Straten, Duco )
• C05 - Strata and tautological classes for compactifications of Shimura varieties (Project Head Wedhorn, Torsten )

• Applicant Institution: Goethe-Universität Frankfurt am Main
• Co-Applicant Institution: Ruprecht-Karls-Universität Heidelberg; Technische Universität Darmstadt
• Participating University: Johannes Gutenberg-Universität Mainz; Universität Münster
• Spokesperson: Professor Dr. Jakob Stix