Mappings between Riemannian manifolds of Sobolev type regularity arise naturally in problems of calculus of variations and partial differential equations motivated by physical models and geometric problems. Because of their nonlinear character, Sobolev spaces of mappings between manifolds have notable differences with the classical linear Sobolev spaces. For instance, topological quantitative and qualitative obstructions arise recurrently and nonlinear constructions are required to prove their properties. Typical strategies involve avoiding singularities by suitable composition on the domain that effectively reduces the dimension, propagating values on higher dimensional sets and bringing back some values onto the target manifold by a suitable projection. This course will start from the approximation, extension and lifting problem for Sobolev maps, develop suitable nonlinear tools for their study and apply the latter to the solution of the problems. The approach will start from the perspective of mathematical analysis and will bring naturally some methods and concepts of homotopy theory for Riemannian manifolds.

Rough path theory provides a fundamental explanation for various approximation methods to solving (stochastic) dynamic systems.  By identifying signature as the key object to introduce and to study a family of metrics on the space of paths. Rough path theory provides a robust solution to controlled differential equations driven by irregular signals which can be applied to Brownian motion, continuous semi-martingales, a large family of Gaussian processes and Markov processes.

PDE-CDT Hilary 2019 Schedule