The doctoral researcher will work at the interface of arrangements of hyperplanes, aperiodic tilings, and the classification of B-free lattice systems. One focus lies on high-dimensional tilings with a focus on aperiodicity and their Heesch numbers. Further questions pertain to infinite simplicial arrangements and their periodic structures. Another focus can be the algebraic and spectral structures of B-free lattice systems.

The position is particularly suited to candidates with a background in algebra, combinatorics or tilings, preferably with experience in some computer algebra system or programming language.

This project lies in geometric approximate group theory - the recent extension of geometric group theory that generalizes concepts and theorems from groups to approximate groups. It studies existence or non-existence of approximate lattices in totally disconnected, locally compact groups (almost) acting on trees that are known to not have any lattices.

The PhD project will be part of a multifaceted research project at the interface of

1) the theory of discrete subgroups in higher rank Lie groups

2) algebro-geometric description of character varieties and their hybrid compactifications

3) spectral analysis on locally symmetric spaces

The position is particularly suited for candidates with background in one of these topics. As a strict requirement, however, we only expect the candidate to be curious to get in touch with all three topics.

The PhD project will be part of a multifaceted research project that investigates the large scale geometry of symmetric spaces and Euclidean buildings, regularity properties of quasi-isometric embeddings between such spaces, as well as related classes of discrete subgroups of Lie groups. The project will be pursued in a collaboration of researchers at the MPI MiS in Leipzig and at Paderborn University, and the PhD candidate will interact with professors and postdocs at both institutions.

This project lies at the interface of representation theory and the combinatorics of cluster algebras. A central theme is the Caldero–Chapoton map, which associates functions to representations of an algebra and in many cases provides an explicit link between finite-dimensional algebras and cluster algebras. The project will investigate the algebras generated by these functions, aiming to understand what cluster-like structure they carry. A related direction concerns distinguished bases of cluster algebras, and how these can be extended to cluster algebras coming from orbifolds. The precise focus will be shaped by the interests and background of the successful candidate, with scope for combining algebraic, geometric and combinatorial methods.

The PhD project is about invariants of triangulated categories that appear in commutative algebra and representation theory, like the derived category or the stable module category, and how they interact with properties of the rings or algebras. The invariants are generation time, which generalizes, among other things, projective dimension, and the lattice of thick subcategories.

The postdoctoral researcher will work at the interface of non-Archimedean geometry, tropical geometry, intersection theory, and matroid theory. The main focus will be the development of non-Archimedean volume polynomials and intersection-theoretic invariants associated with metrised vector bundles, with applications to the geometry and Hodge theory of valuated matroids. Toric varieties and toric schemes will provide an important testing ground for these ideas.

The position is particularly suited to candidates with a strong background in non-Archimedean or tropical geometry, intersection theory, or matroid theory.

In Project A4 we investigate combinatorial Euler products arising from a variety of enumeration problems in algebra, geometry, and number theory. These include the enumeration of finite-index subgroups or finite-dimensional representations of infinite (often arithmetic or profinite) groups, orders in rings of integers of number fields, or finite extensions of global or local fields. Often the solutions to these problems are best recorded through suitably defined generating series, or zeta functions. These are then studied with a range of algebraic, analytic, and combinatorial methods.

The Bielefeld postdoc position is particularly suited to candidates with a background in asymptotic questions in group or ring theory, algebraic or analytic number theory, or combinatorial algebra.

In project A5, we investigate Kac–Moody groups from various perspectives. One aspect of interest is groups of integer points in affine Kac–Moody groups, i.e., groups like $\mathrm{SL}_n( \mathbf{Z}[t,t^{-1}] )$. We will be investigating finiteness properties of these groups (finite generation, finite presentability, and higher finiteness properties).

Project B4 is concerned with filling functions and isoperimetric inequalities for arithmetic groups in positive characteristic and consequently with measures of distortion for subspaces of Euclidean buildings. It turns out that filling functions are related to questions about "how much of a flat is close to infinity". Hence, this project connects geometry, spectral theory and tools from probability.

The postdoctoral researcher will work at the interface of logarithmic and tropical geometry, combinatorics, locally symmetric varieties, and automorphic forms. The main focus will be the construction and study of logarithmic compactifications of locally symmetric varieties, together with the development of logarithmic tautological rings generated by special cycles, characteristic classes of automorphic vector bundles, and boundary classes. A further direction concerns the combinatorial structures underlying these compactifications and their cohomology, including connections to complexes of regular matroids.

The position is particularly suited to candidates with a strong background in logarithmic or tropical geometry, combinatorial algebraic geometry, the geometry of locally symmetric varieties, or automorphic forms.

The postdoctoral researcher will be part of a multifaceted research project that investigates the large scale geometry of symmetric spaces and Euclidean buildings, regularity properties of quasi-isometric embeddings between such spaces, as well as related classes of discrete subgroups of Lie groups. The project will be pursued in a collaboration of researchers at the MPI MiS in Leipzig and at Paderborn University, and the postdoctoral researcher will interact with professors and postdocs at both institutions.