Talks and abstracts

Topics for the main lectures

Karen Habermann: Cartan connections for stochastic developments on sub-Riemannian manifolds

Abstract: Analogous to the characterisation of Brownian motion on a Riemannian manifold as the development of Brownian motion on a Euclidean space, we construct sub-Riemannian diffusions on equinilpotentisable sub-Riemannian manifolds by developing a canonical stochastic process arising as the lift of Brownian motion to an associated model space. The notion of stochastic development we introduce for equinilpotentisable sub-Riemannian manifolds uses Cartan connections, which take the place of the Levi-Civita connection in Riemannian geometry. We derive a general expression for the generator of the stochastic process which is the stochastic development with respect to a Cartan connection of the lift of Brownian motion to the model space. For free sub-Riemannian structures with two generators, we illustrate the construction of a suitable Cartan connection which develops the canonical stochastic process to the sub-Riemannian diffusion associated with the sub-Laplacian defined with respect to the Popp volume. This is joint work with Ivan Beschastnyi and Alexandr Medvedev.

Ben McKay: Holomorphic Cartan geometries

Abstract: What are the prettiest geometries on the prettiest manifolds? I will review some complex algebraic geometry, the known holomorphic parabolic geometries on smooth projective varieties, the conjectured classification, and how much we can prove of it so far.

The first lecture is an introduction to general Cartan geometries. Assuming notions of manifold and Lie group, I will motivate and explain the definition of Cartan geometry, and discuss developing of curves, automorphism groups, homogeneous geometries, and invariant pseudometrics. The following two lectures will be focused on holomorphic Cartan geometries.

Josef Šilhan: Tractor calculus in conformal and projective geometry

Abstract: Firstly, we introduce conformal and projective geometry and tractor calculus on these structures. Secondly, we discuss compatible (or invariant) families of overdetermined systems of PDE’s and show how to find their prolongation via tractors. Thirdly, we show how to build a tractor analogue of Frenet frames for curves on these geometrical structures.