# Faculty of Mathematics

### Algebraic Geometry and Arithmetic

The research topics of the Essen Seminar for Algebraic Geometry and Arithmetic (ESAGA) centre on Algebraic Geometry, Number Theory and Algebraic Topology. Motivic homotopy theory, algebraic cobordism, questions about rational points, the classical and the p-adic Langlands program, Shimura varieties, lattices and theta series, algebraic stacks, moduli spaces, classification theory of complex algebraic varieties, L-functions, and the representation theory of p-adic Lie groups are among the specific research interests of its members.

The many connections between these subjects lead to an intense exchange between the research groups of the Essen Seminar. The Transregio Collaborative Research Centre 45 “Moduli Spaces, Periods, and Arithmetic of Algebraic Varieties” (Mainz/Bonn/Essen) was extended in 2015 for the second time and has now entered its third 4-year period. All the eight research groups were successful with their project proposals. A great success, all the more considering that all the professors in the field of algebra were hired after the SFB’s beginning in 2007. The CRC enables the Essen Seminar to carry out a multitude of activities, includes a good number of positions for PhD students and post-docs, and immensely increases the possibilities of visiting mathematicians elsewhere, and of inviting them to Essen.

The Priority Programme 1786 “Homotopy Theory and Algebraic Geometry” of the German Research Foundation, which was initiated and is now coordinated by Prof. Marc Levine, began in May 2015. In this programme, new developments connecting algebraic topology and algebraic geometry are investigated at universities all over Germany. Since its start, the programme has already seen a large number of conferences and workshops, among them several taking place in Essen. Prof. Ulrich Görtz participated with a project in the Priority Programme 1489 “Algorithmic and Experimental methods in Algebra, Geometry and Number Theory” which ran until 2016.Since October 2014, Essen has been one of the host locations for the “Joint Seminar on Complex Algebraic Geometry and Complex Analysis”, which is organized by the complex geometry research groups at the Universities of Bochum, Duisburg-Essen (research group Greb), Cologne, and Wuppertal. The seminar meets twice per semester for a full day at one of the participating universities. It is a platform for talks both by invited guests (among others from the Universities of Bonn, Moscow, Stony Brook, and Zurich) and by speakers from the organizing universities, as well as for discussions among the members of the research groups involved. It has already proven to be a perfect base for developing collaborative research projects.

As usual, the Essen seminar welcomed a large number of guests from abroad, some of whom spent several months in Essen, among them Gergely Zábrádi (Eötvös Loránd University, Budapest), Henri Darmon (McGill University, Montreal) and Marco Seveso (Università degli Studi di Milano). Members of the Essen Seminar collaborated with other mathematicians from all over the world, for instance with Pierre Colmez (CNRS Paris), Matthew Emerton (University of Chicago), Oscar Garcia-Prada (CSIC Madrid), Toby Gee (Imperial College, London), Tamas Hausel (EPFL Lausanne), Xuhua He (University of Maryland), Julius Ross (Cambridge), and Jerzy Weyman (University of Connecticut). At our university, a school on p-adic arithmetic with around 75 participants from Europe and North America took place in March 2015. In September 2016, a school on Higgs bundles and fundamental groups of algebraic varieties was organized with approximately 45 participants from Europe and India.