Deformations, stability and rigidity in higher Lie theory

• Applicant: Dr. Karandeep Jandu Singh
• Subject Area: Mathematics
• Term: since 2024
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 549348984

Differential geometry is the study of smooth spaces called manifolds. Examples of manifolds include the plane, the 3-dimensional space around us, a circle and the surface of a ball. The notion of smoothness is not well-behaved under natural operations of spaces: for instance, gluing, which is the process of identifying different points and is necessary in the presence of symmetries, does not preserve smoothness. To remedy this, various approaches exist, one of them being the language of Lie algebroids and Lie groupoids. Roughly speaking, instead of the glued space, Lie groupoids keep track of the separate spaces, and the way they are glued together. Lie algebroids are a linear approximation to Lie groupoids, and the gluing data can be recovered from the Lie algebroid. Hence, constructions on Lie groupoids and Lie algebroids encode constructions on the glued space. In this project, we will study the deformation theory of Lie algebroids and their generalizations. By studying what properties are preserved under small changes, we aim to get a better understanding of the space of Lie algebroids, which is infinite-dimensional and highly singular. We will study both concrete questions, and the general theory. As concrete questions, we will study whether specific algebraic or dynamical properties are preserved under small changes, as well as construct the algebraic structure underlying the deformation theory of double Lie algebroids. For the general theory, we will place the concrete questions in a more general framework, using the algebraic structure underlying the deformation theory of the respective structure, and investigate whether sufficient conditions can be obtained in the language of the general algebraic structure.

• DFG Programme: WBP Position