The Hypoelliptic Laplacian in Real and Complex Geometry

Explore the construction and applications of the hypoelliptic Laplacian in real and complex geometry through this 56-minute lecture by Jean-Michel Bismut from Université Paris-Saclay, France. Delve into the deformation of the Laplacian by a family of operators acting on a larger space, often the total space of the tangent bundle, which interpolates between the Laplacian and the generator of the geodesic flow. Understand the geometric Fokker-Planck operators and their specific geometric content in different contexts. Discover the motivations behind this construction, including the interpolation property and the ability to lift geometric obstructions such as the existence of a Kähler metric. Examine the hypoelliptic Laplacian's implementation in de Rham theory and Dolbeault theory, gaining insights into this advanced mathematical concept and its implications for real and complex geometry.