TRR 45:  Periods, Moduli Spaces and Arithmetic of Algebraic Varieties

• Subject Area: Mathematics
• Term: from 2007 to 2019
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 30164218

Periods, moduli spaces and related aspects of the arithmetic of algebraic varieties are central research themes inside the SFB/Transregio 45. These topics and some of the most exciting recent developments around them touch central questions in the intersection of arithmetic and geometry. Although algebraic geometry and number theory have coexisted for a long time, the close connection between these two areas that we now enjoy began only in the twentieth century. The development of schemes and general cohomology theories, as well as the theory of motives, provided a common framework for both. A prominent example for such interplay is Wiles' proof of the Fermat conjecture, which uses the geometry of elliptic curves and their Tate modules to prove a genuinely arithmetic statement. In our research we aim to develop further tools relating arithmetic and geometry, and hope to build as yet unknown bridges between the two different points of view. We have chosen a selected group of PIs with similar common research interests so that a close interaction would be possible. Our main goal besides advancing our understanding of this interplay is to motivate young mathematicians to enter this fascinating area of research and enable them to see the geometric and arithmetic perspectives from the beginning of their studies.Many of the main research topics have been carried over from the first and second funding period. In addition, we have added several new research topics, e.g., p-adic representations and L-functions, local Shimura varieties and diophantine problems, cubic 4-folds and related questions, geometric representation theory, Quiver Schur Algebras, Jacobian Algebras, K-theory of stacks, Mumford-Tate domains, elliptic motives, several aspects of birational geometry and the geometric Langlands correspondence.

• DFG Programme: CRC/Transregios
• International Connection: Vietnam

Completed projects

• A01 - Periods of the nilpotent completion of the fundamental group (Project Heads Esnault, Hélène ;  Phùng, Hô Hai )
• A02 - Tannaka group schemes of certain categories of bundles (Project Heads Esnault, Hélène ;  Phùng, Hô Hai )
• A03 - Higgs bundles and Higgs cohomology on quasi-projective manifolds (Project Head Zuo, Kang )
• A04 - Feynman integrals and motives (Project Head Esnault, Hélène )
• A05 - Some aspects of limiting mixed Hodge structures (Project Heads Esnault, Hélène ;  Viehweg, Eckart )
• A06 - Motivic cycles and regulators (Project Head Müller-Stach, Stefan )
• A07 - Modular Galois representations and Galois theoretic lifts (Project Head Böckle, Gebhard )
• A08 - Universal deformations, the regidity method and Galois representations (Project Head Böckle, Gebhard )
• A09 - Arithmetic of Katz modular forms (Project Head Wiese, Gabor )
• A10 - The cohomology of A-crystals, moduli spaces in positive characteristic and p-adic étale cohomology on schemes over Z_p (Project Head Böckle, Gebhard )
• A12 - Congruences for the number of rational points over finite fields (Project Head Esnault, Hélène )
• A13 - p-adic point counting on Calabi-Yau threefolds (Project Head van Straten, Duco )
• B02 - Period domains of hyperkähler manifolds (Project Head Huybrechts, Daniel )
• B03 - Picard-Fuchs equations, monodromy, and the Mumford-Tate group of special families of CY manifolds (Project Head Müller-Stach, Stefan )
• B04 - Picard-Fuchs equations of Calabi-Yau type (Project Head van Straten, Duco )
• B06 - Periods and period domains for Abelian varieties (Project Heads Müller-Stach, Stefan ;  Viehweg, Eckart ;  Zuo, Kang )
• B07 - Local models of Shimura varieties (Project Head Rapoport, Michael )
• B08 - Affine Deligne-Lusztig varieties (Project Head Görtz, Ulrich )
• B09 - SL_2(IR)-action on translation surfaces and Techmüller curves (Project Head Möller, Martin )
• B11 - Special subvarieties of Shimura varieties (Project Head Zuo, Kang )
• C01 - Algebraic Calabi-Yau categories (Project Head Schröer, Jan )
• C03 - Derived categories of Calabi-Yau manifolds (Project Head Huybrechts, Daniel )
• C04 - Nonliftable Calabi-Yau manifolds in positive characteristics (Project Head van Straten, Duco )
• C05 - Lagrangian fibrations of symplectic manifolds (Project Heads Huybrechts, Daniel ;  Lehn, Manfred )
• C06 - Rozansky-Witten anvariants (Project Head Nieper-Wißkirchen, Marc )
• C07 - Symplectic Singularities (Project Head Lehn, Manfred )
• C10 - Moduli with GIT (Project Head Schmitt, Alexander )
• C11 - Construction of moduli spaces: compactifications and ample sheaves (Project Heads Esnault, Hélène ;  Viehweg, Eckart )
• M01 - Fundamental Groups (Project Heads Blickle, Ph.D., Manuel ;  Esnault, Hélène ;  Müller-Stach, Stefan ;  Zuo, Kang )
• M02 - Periods and Motives (Project Heads Esnault, Hélène ;  Geldhauser, Nikita ;  Kerz, Moritz ;  Levine, Marc ;  Müller-Stach, Stefan ;  Stroppel, Catharina )
• M03 - Galois Representations, Finite and Mixed Characteristics (Project Heads Blickle, Ph.D., Manuel ;  Faltings, Gerd ;  Görtz, Ulrich ;  Heinloth, Jochen ;  Paskunas, Vytautas ;  Scholze, Peter ;  van Straten, Duco ;  Wiese, Gabor ;  Zuo, Kang )
• M03 - Ramification in the geometric Langlands correspondence The Breuil-Mezard conjecture (Project Heads Heinloth, Jochen ;  Paskunas, Vytautas )
• M04 - Rational Points (Project Heads Bertolini, Massimo ;  Esnault, Hélène ;  Faltings, Gerd ;  Levine, Marc )
• M05 - Periods and Period Domains (Project Heads Faltings, Gerd ;  Kohlhaase, Jan ;  Müller-Stach, Stefan ;  Rapoport, Michael ;  Scholze, Peter ;  van Straten, Duco ;  Zuo, Kang )
• M06 - Shimura Varieties (Project Heads Bertolini, Massimo ;  Görtz, Ulrich ;  Hellmann, Eugen ;  Müller-Stach, Stefan ;  Paskunas, Vytautas ;  Rapoport, Michael ;  Scholze, Peter ;  Stroppel, Catharina ;  Viehmann, Eva ;  Zuo, Kang )
• M06 - Completed cohomology (Project Head Paskunas, Vytautas )
• M07 - Calabi-Yau Categories (Project Heads Huybrechts, Daniel ;  Macri, Emanuele ;  Schröer, Jan )
• M08 - Moduli Spaces, Symplectic and Calabi-Yau Manifolds (Project Heads Greb, Daniel ;  Hein, Georg ;  Heinloth, Jochen ;  Huybrechts, Daniel ;  Lazic, Vladimir ;  Lehn, Manfred ;  Rollenske, Sönke ;  van Straten, Duco ;  Stroppel, Catharina )
• M08 - Stability conditions on moduli stacks and applications to the geometry of moduli spaces (Project Head Heinloth, Jochen )
• MGKG - Integrated Research Training Group (IRTG) (Project Heads Blickle, Ph.D., Manuel ;  Müller-Stach, Stefan )
• Z - Administrative Project (Project Heads Blickle, Ph.D., Manuel ;  Müller-Stach, Stefan )

• Applicant Institution: Johannes Gutenberg-Universität Mainz
• Co-Applicant Institution: Rheinische Friedrich-Wilhelms-Universität Bonn; Universität Duisburg-Essen; Campus Essen (aufgelöst)
• Participating Institution: Max-Planck-Institut für Mathematik
• Spokespersons: Professor Manuel Blickle, Ph.D., since 1/2017; Professor Dr. Stefan Müller-Stach, until 12/2016