# Geometry: Deformations and Rigidity

From its historic roots, geometry has evolved into a central subject in modern mathematics, both as a tool and as a subject in its own right. The research programme of the CRC proposes to approach a variety of mathematical problems geometrically from two seemingly antagonistic but complementary poles: Deformations and Rigidity.

Deformations of mathematical objects can be viewed as continuous families of these. All possible deformations of a mathematical object can often be considered as a deformation space or a moduli space. The geometric properties of this space in turn shed light on the deeper structure of the given mathematical object.

Conversely, a rigidity phenomenon refers to a situation where essentially no deformations are possible: Properties or quantities associated with mathematical objects are rigid if they are preserved under all reasonable deformations.

The dichotomy of deformations and rigidity appears in the study of various geometric contexts in mathematics, notably in the Langlands programme, positive curvature manifolds, partial differential equations, K-theory, group theory, and C∗-algebras. These research directions are the cornerstones of the research in our CRC.

The unifying principle of deformations and rigidity led us to structure our research via five overarching research objectives: Arithmetic, K-Theory, Moduli Spaces, Curvature and Groups and Dynamics. These research objectives serve as guiding ideas for multiple projects by tying together concrete questions.