GRK 2965:  From geometry to numbers: Moduli, Hodge theory, rational points

• Subject Area: Mathematics
• Term: since 2024
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 512730679

An important phenomenon of mathematics is the interplay between geometry and numbers. A first manifestation of this idea is the fact that geometric objects can be assigned algebraic invariants and, in particular, numbers, which often play an important role in classification. Examples are given by Betti and Hodge numbers, by intersection numbers on moduli spaces, by Mordell-Weil ranks and by the number of rational points. Another important connection is given by the fact that many fundamental questions in number theory, such as the solvability of a Diophantine equation, naturally describe geometric figures whose properties control the underlying arithmetic problems. Conversely, various arithmetic ideas and methods have important applications in geometry, and in particular in Hodge theory and moduli theory. Our scientific program focuses on A) Hodge theory and topology of algebraic varieties, B) geometry and combinatorics of moduli, and C) arithmetic of moduli and rational points. These areas are closely intertwined. Advances in one area often incorporate ideas, methods, or results from one of the other fields. Typical examples of the interplay of all three areas are recent breakthroughs concerning the distribution of Hodge loci and on effective versions of Faltings' theorem. Progress on questions about moduli have led to remarkable results in the topology and Hodge theory of algebraic varieties. At the same time, techniques from Hodge theory and sheaf theory have important applications in the moduli theory of curves, surfaces, and abelian varieties. Together with the joint expertise of the participating groups at Humboldt-Universität zu Berlin and Leibniz Universität Hannover, these links provide the framework for our graduate school. The goal is to combine our methods, results and expertise to address fundamental problems in all three areas mentioned above. Our research training group offers an environment for ambitious PhD projects and a broad, yet focused, qualification program that prepares the PhD students step by step to become independent researchers. Our supervision strategy is based on close collaboration between the two sites, usually with a PhD supervisor from one site and a co-advisor from the other, and relies on a variety of joint activities. Our goal is to provide our PhD students with an inspiring environment for exceptional performance during their PhD in the stimulating context of two vibrant and internationally visible research groups in Berlin and Hannover. Together, we are committed to providing our students with excellent prospects for a promising career in or outside academia.

• DFG Programme: Research Training Groups
• Applicant Institution: Gottfried Wilhelm Leibniz Universität Hannover
• Co-Applicant Institution: Humboldt-Universität zu Berlin
• Spokesperson: Professor Dr. Stefan Schreieder
• Participating Researchers: Professor Dr. Gaëtan Borot; Professor Dr. Michael Cuntz; Professor Dr. Ulrich Derenthal; Professor Dr. Gavril Farkas; Professor Dr. Ziyang Gao; Professor Dr. Bruno Klingler; Professor Dr. Thomas Krämer; Privatdozentin Dr. Angela Ortega; Professor Dr. Matthias Schütt; Professorin Dr. Isabel Stenger