# Algebraic Geometry and Arithmetic: the research training group ’symmetries and Classifying Spaces’

This year, the research training group ‘symmetries and Classifying Spaces: Analytic, Arithmetic and Derived’ was established at our faculty. Its research groups from the Essen Seminar for Algebraic Geometry and Arithmetic aim to capitalise on the numerous interfaces of their research projects for doctoral training in an extraordinarily dynamic field. The group leaders are Massimo Bertolini, Ulrich Görtz, Daniel Greb, Georg Hein, Jochen Heinloth, Jan Kohlhaase, Marc Levine, Ursula Ludwig, Andreas Nickel and Vytautas Paskunas.

Symmetries and the classification of geometric objects are key issues in mathematics and, in particular, the various approaches to algebraic geometry: classical algebraic geometry, complex geometry, arithmetic geometry, derived algebraic geometry and other fields at the intersection of algebraic geometry, analysis and topology. Our research focuses on groups that describe symmetries of geometric, analytic and number-theoretic objects and classifying spaces, i.e., spaces that parametrise all objects of a given type, to varying degrees. The two topics are often closely connected.

Enormous progress has been made over the past years: the theory of perfectoid spaces, the Langlands programme, progress on the Birch and Swinnerton-Dyer conjecture and the minimal-model programme are famous examples. The development of new methods allows the field to advance rapidly, and new breakthroughs are on the horizon. This makes it a promising field of research for young mathematicians looking to start their careers. With a multitude of methods in use, doctoral candidates benefit from working in an environment where they have access to expertise in many of the numerous approaches. The Faculty of Mathematics offers such a stimulating environment to its early-career researchers in Essen. Our doctoral candidates are supported in their transitions from students to researchers and get to establish themselves in a fascinating mathematical field.

We will outline some of the results produced by the research groups who have contributed to this project and emphasise the interfaces between them. The construction of integer or rational solutions to equations is a fundamental problem of number theory. The Birch and Swinnerton-Dyer conjecture describes a relationship between the structure of rational solutions and the analytically defined invariants of the underlying properties. In order to understand that mysterious relationship, it seems inevitable that we must develop methods for constructing arithmetic solutions using analytical objects. Professor Bertolini’s research group has proven new results of this type for p-adic L-functions.

Professor Paskunas’s group was also able to use p-adic methods to produce new global, arithmetic results, also taking advantage of the local geometry of classifying spaces. These classifying spaces formally resemble the spaces of spaces of representations of groups arising in geometry, which the research groups of Professors Greb, Hein and Heinloth study. In both situations, the same issue occurs: the global geometry of the problems frequently exhibits pathologies which lead to stability conditions for the objects parametrized by these spaces. Surprisingly, analytic descriptions, which may often be formulated in terms of stability results for solutions of certain differential equations, also yield conditions that can be understood in purely algebraic terms. To understand the relationship between these stability conditions and the geometry of the parameter spaces, the researchers were able to prove results that allow them to reduce the study of stability properties to a few key conditions. Because the spaces thus obtained often exhibit singularities, it is difficult to approach them with analytic methods. Dr Ursula Ludwig’s work focuses on this obstacle and expands it to include fundamental analytical methods for interesting classes of singularities.