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# Research Training Group 2553

The Research Training Group 2553 “Symmetries and classifying spaces: analytic, arithmetic and derived” was approved by the German Research Foundation in November 2019 and started in April 2020.

**Summary (from our proposal)**

The study of symmetries and classification of geometric objects lies at the heart of mathematics and of algebraic geometry in all of its flavors: classical algebraic geometry, complex geometry, arithmetic geometry, derived algebraic geometry and other areas at the border between algebraic geometry, analysis and topology. This is a very active subject that has made substantial progress in the past years, e.g., the theory of perfectoid spaces, a better understanding of the Langlands programme and of the Birch and Swinnerton-Dyer conjecture, and new results on the minimal model programme. New tools are developing quickly and further break-throughs can be expected in the future. For young mathematicians this is a promising field in which to start one’s career. In view of the difficulty and the breadth of the methods, it is particularly useful for PhD students to study and work in an environment where expertise in many of the different approaches to the subject is available. At Essen, we can offer such a stimulating environment, supporting PhD students in the transitional phase between student and researcher, and enabling them to enter a fascinating area of mathematics.

Our research will focus on groups and classifying spaces in a broad sense, fruitfully combining two closely related topics. This includes complex and $p$-adic Lie groups, algebraic groups and Galois groups, their actions on varieties and related spaces and representation theoretic questions with a link to geometric problems, as well as moduli spaces, deformation spaces, and classifying spaces in the strict sense. Symmetries and classifying spaces are often closely linked: moduli spaces are frequently constructed as quotients; the objects being parameterized might carry a group structure (abelian varieties) or be related to a group ($G$-bundles); cohomology of moduli spaces provides interesting representations.

In the qualification programme, our goal is to provide students with enough freedom to develop their own ideas and become, step by step, more independent, and to provide enough guidance to promote an effective use of their time and to ensure their work has a significant perspective going beyond the PhD thesis, in an environment that allows them to concentrate fully on mathematics.

**Core topics of the Research Training Group:**

- Moduli Spaces and Deformation Spaces
- Lie Groups, their Actions, and Quotient Spaces
- Duality
- Galois and Automorphic Representations

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