Geometry: Deformations and Rigidity

From its historic roots, geometry has evolved into a cornerstone in modern mathematics, both as a tool and as a subject in its own right. On the one hand many of the most important open questions in mathematics are of geometric origin, asking for example to what extent an object is determined by geometric properties. On the other hand, abstract mathematical problems can often be solved by associating them to more geometric objects that can then be investigated using geometric tools. A geometric point of view on an abstract mathematical problem quite often opens a path to its solution.

Deformations and rigidity are two antagonistic geometric concepts which can be applied in many abstract situations making transfer of methods particularly fruitful. Deformations of mathematical objects can be viewed as continuous families of such objects, like for instance evolutions of a shape or a system with time. The collection of all possible deformations of a mathematical object can often be considered as a deformation space (or moduli space), thus becoming a geometric object in its own right. The geometric properties of this space in turn shed light on the deeper structure of the given mathematical objects. We think of properties or of quantities associated with mathematical objects as rigid if they are preserved under all (reasonable) deformations.
A rigidity phenomenon refers to a situation where essentially no deformations are possible.
Rigidity then implies that objects which are approximately the same must in fact be equal, making such results important for classifications.

The overall objective of our research programme can be summarised as follows:

Develop geometry as a subject and as a powerful tool in theoretical mathematics focusing on the dichotomy of deformations versus rigidity. Use this unifying perspective to transfer deep methods and insights between different mathematical subjects to obtain scientific breakthroughs, for example concerning the Langlands programme, positive curvature manifolds, K-theory, group theory, and C*-algebras.