Geometric Structures Laboratory

The role of Lie theory in classification problems in geometry goes all the way back to Cartan and even Lie himself. Nowadays, however, the languages of Lie theory and differential geometry have somewhat diverged. Bryant observed that underlying certain classification problems there is a Lie algebroid. Fernandes and Struchiner exploited this remark and described the precise role of the integration of the Lie algebroid in finding the global solutions to a geometric problem. However, their theory only applies to problems whose local classification is finite-dimensional.

Relative algebroids were introduced as the Lie theory behind generic geometric problems. In this mini-course, I will explain, through examples, the foundations of relative algebroids, their prolongations and formal integrability, ending with an explicit example to which the framework is applied. If there is time, I will also explain how they can be used to study global aspects of PDEs with symmetry.