TRR45 Collaborative Research Center

The TRR45 Peridods, Moduli spaces and the Arithmetik of Algebraic Varieties is a collaborative research center among the mathematical institutes of Bonn, Essen and Mainz. This conference is to celebrate the successful 12 years of mathematical research at the border of algebraic and arithmetic geometry within the SFB/TRR45.

Collaborative Research Centers (CRC) of the German Research Council exist for more than 50 years. The main feature of a CRC is a focus on a strong and active field of research, usually defined by a concrete description in its title, and a strong group of roughly 20 principal investigators who work in that area and all show a strong past performance in the field.

In 2007, TRR45 “Periods, moduli spaces, and arithmetic of algebraic varieties”, a transregional CRC based in Bonn, Essen and Mainz was founded.  It combines three strong research groups in arithmetic and algebraic geometry at the three universities. Many award winning PIs were part of TRR45, including two field medal awardees.  Many young academics were part of the TRR at an early stage of their careers and later got appointed at other universities. There were many lively and and stimulating mathematical activities, such as our summer/winter schools and workshops/conferences which benefitted the immersion of our doctoral students in this deep and fascinating research area.

The research area of TRR45 was designed to be at the intersection of arithmetic and algebraic geometry with a strong focus on the topics of periods and moduli spaces. During the first funding period, the research projects were grouped into the core ares (A) Motives and Periods, (B) Period Domains and (C) Moduli Spaces of Special Varieties. For the remaining funding periods the key research areas where several PI’s closely worked together were:

  • Fundamental Groups
  • Motives and Periods
  • Galois Representations
  • Rational Points
  • Periods and Period Domains
  • Shimura Varieties
  • Calabi-Yau Categories
  • Moduli Spaces, Symplectic and Calabi-Yau Manifolds

Some of the biggest successes in research were of course developed by Peter Scholze who became part of the CRC during the third funding period. His development of perfectoid geometry was a hallmark in mathematics, and his participation in projects of the TRR defined many aspects of the field of arithmetic and algebraic geometry in an entirely new way. However, there were many other fascinating subjects which lead to new and unexpected results. This list contains in particular the study of periods and period domains (hence Shimura varieties), the theory of motives and motivic homotopy theory, the study of Galois representations, the theory of varieties in finite and mixed characteristics, p-adic geometry, and the study of different aspects of special varieties, in particular of symplectic and Calabi-Yau type. As a whole, the research inside TRR45 and the impact it had on  the careers of many young researchers was tremendous.

During the 12 years of its tenure the following PI’s participated in the SFB/TRR45, vaguely listed in chronological order.

Gebhard Böckle, Hélène Esnault, Gerd Faltings, Ulrich Goertz, Daniel Huybrechts, Manfred Lehn, Marc Levine, Stefan Müller-Stach, Michael Rapoport, Jan Schröer, Duco van Straten, Eckart Viehweg, Kang Zuo Martin Möller, Marc Nieper-Wisskirchen, Ho-Hai Phung, Alexandrea Sarti, Alexander Schmitt, Gabor Wiese, Moritz Kerz, Emanuele Macri, Sönke Rollenske, Manuel Blickle, Nikita Semenov, Catarina Stroppel, Massimo Bertolini,  Jochen Heinloth. Georg Hein, Eugen Hellmann, Jan Kohlhase,  Vitautas Paskunas, Peter Scholze.