Invariant involutive structures

• Applicant: Dr. Max Jahnke
• Subject Area: Mathematics
• Term: since 2024
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 547454521

It is well known that the De Rham cohomology of a compact Lie group is isomorphic to the Chevalley-Eilenberg cohomology. While the former is a topological invariant of the Lie group, the latter can be computed by using simple linear algebra methods. Our main goal is to understand if it is possible to obtain a similar isomorphism between the cohomology spaces associated with invariant involutive structures and the cohomology of a generalized Chevalley-Eilenberg complex. This project proposes a highly collaborative approach, bringing together experts in CR Geometry, Global Analysis, Several Complex Variables and Partial Differential Equations, to achieve this and related goals. In particular, by combining the applicant's expertise these areas with techniques from Algebraic Topology and Lie theory, we believe that it is possible to obtain new exciting results in the cases of elliptic, CR, and essentially real involutive structures defined over a compact Lie group or a compact nilmanifold as well as local and global solvability for certain CR structures. The results will provide new insights in the general theory of involutive structures while also revealing algebraic obstructions for solvability for the associated differential complexes on compact Lie groups.

• DFG Programme: Research Grants