The asymptotic viewpoint in geometry provides a common framework for discrete and continuous spaces and underlies our research program. The research program is structured into three core research areas:
A Asymptotic invariants of groups and spaces
- A1 Higher order Dehn functions
- A2 Boundaries
- A3 ℓ2-Invariants
B Deformation and moduli spaces
- B1 Geometric structures on infinite surfaces
- B2 Flows and parametrizations of deformation spaces
- B3 Moduli spaces of Riemannian metrics and the Willmore functional on moduli spaces of curves
C Converengence, limits and degenerations of spaces
- C1 Solutions of geometric partial differential equations
- C2 Compactifications
- C3 Invariant random subgroups and notions of convergence
Core research idea
The core research idea of the proposed RTG is the systematic study of geometric spaces with regard to their asymptotic invariants and their behavior under deformations, degenerations, and taking limits.
The methods are drawn from differential geometry, geometric analysis, geometric group theory, and topology. Several breakthroughs in recent years reveal a fascinating picture of mutual influence between these areas. The most famous examples are Perelman’s proof of the Poincare conjecture, a topological problem solved by Riemannian geometry and geometric analysis, and Agol’s and Wise’s proof of Thurston’s virtual Haken conjecture, a topological problem solved by geometric group theory.
The asymptotic or large-scale viewpoint, which focuses on macroscopic properties and ignores the local geometry, enables a unified treatment of diverse geometric objects, such as Riemannian manifolds, metric measure spaces, and infinite groups (via the word-metric). It underlies many interactions of the aforementioned areas. Under the asymptotic viewpoint, the classification of geometric objects with regard to their large-scale geometry is a fundamental problem. Invariants, which uncover key properties of mathematical objects, are often a step towards a classification. The most classical approach to classifying geometrical objects is by numerical invariants such as dimension, genus, or Euler characteristic. However, one cannot adequately capture asymptotic properties by these classical invariants. To mend this, we will consider asymptotic invariants of groups and spaces (Research area A) instead. As an example from our research agenda, we mention ℓ2-Betti numbers. In recent years, research activity surged with regard to generalizations of the approximation theorem for ℓ2-Betti numbers, opening up new directions we want to pursue.
As geometric spaces often arise in continuous families, we are naturally led to investigate their deformation and moduli spaces (Research area B). These moduli spaces can be endowed themselves with a topological or geometrical structure. Considering the geometric objects now as points of a topological space, special deformations or flows can be defined to relate them to each other. Analyzing the behavior of invariants or functionals over the moduli spaces becomes a crucial question. The investigation of moduli spaces in some cases even leads to new invariants. As for a specific example, we will study higher Teichmüller spaces. They are far-reaching generalizations of classical Teichmüller spaces, the moduli spaces of compact Riemannian surfaces, leading to many new questions, unexpected phenomena, and a growing interaction with theoretical physics.
The moduli or deformation spaces that govern geometric features of manifolds, metric spaces and group actions are inherently non-compact. This makes analyzing convergence properties in general hard. A useful but challenging task is to compactify moduli spaces. Boundary points in these compactifications correspond to limits and degenerations (Research area C) of geometric objects, which form the third aspect of our research program. Apart from limits coming from boundary points, we will also study abstract notions of limits and convergence where no moduli space is around. Examples in our research plan are Benjamini-Schramm limits and limits in the Gromov-Hausdorff topology.