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# Core research topics of the Research Training Group

## Moduli Spaces and Deformation Spaces

The construction and study of the global geometry and deformation theory of moduli spaces parameterizing analytic, algebraic, or arithmetic objects has a rich history and is one of the central objectives in our research programme. The scope of the investigation ranges from fundamental work regarding general criteria for the existence of moduli spaces for algebraic and analytic stacks, over the complex and real geometry of moduli spaces parameterizing Higgs bundles or vector bundles, to moduli spaces occurring in arithmetic questions such as moduli spaces of $p$-divisible groups and Shimura varieties. Quotients of Hermitian symmetric domains are quite generally important objects of study in complex and differential geometry. In addition, Shimura varieties appear here in their capacity as algebro-geometric moduli spaces and owing to their rich interplay with arithmetic applications.

Research topics include:

- Construction of algebraic and analytic moduli spaces
- Geometry of algebraic and analytic moduli spaces
- Derived algebraic geometry and motivic virtual fundamental classes
- Arithmetic applications of moduli spaces: Reciprocity laws
- Shimura varieties: Geometry of Newton strata
- Moduli spaces and universal covers of $p$-divisible groups

## Lie Groups, their Actions, and Quotient Spaces

Lie groups (real, complex, $p$-adic) and linear algebraic groups form an important class of groups with a close tie to geometry, and with many applications for example in the construction and study of moduli spaces. While there are many differences between complex analytic, algebraic geometric and $p$-adic analytic techniques, there is also a large common foundation to these theories, which will play a crucial role in the education of our PhD students.

Research topics include:

- Moment measure conjecture
- Characterisation of compact quotients of Hermitian symmetric spaces
- Analytic and topological invariants of quotient spaces
- Geometry of affine Grassmannians
- Equivariant vector bundles

## Duality

A particular form of symmetry which is ubiquitous in mathematics is duality. Specifically, we intend to study dg categories with duality. In a sense, the various forms of Langlands (and similar) correspondences which are also studied in other projects, or at least are present in the background there, can be seen as a form of duality. More concretely, that formalism involves several instances of dualities, such as the Langlands dual group.

Research topics include:

- dg categories with duality and non-commutative Chow-Witt motives
- Duality and Euler characteristics over a general base
- De Rham theorem for intersection space cohomology
- The $p$-adic Langlands programme and global applications
- $p$-adic $L$-functions and the Drinfeld tower

## Galois and Automorphic Representations

Understanding the structure of Galois groups such as the absolute Galois group of $\mathbb Q$ is one of the main questions which have driven the development of number theory and arithmetic geometry in the last decades. Nowadays the primary approach to achieve this is to understand the representations of such Galois groups. Via the Langlands program, Galois representations are connected with Lie group representations/automorphic representations.

Research topics include:

- Galois representations and elliptic curves over imaginary quadratic fields
- Refined Iwasawa theory and higher Fitting invariants
- Deformation rings of Galois representations
- Representations of $p$-adic Lie groups