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.

In the first half of the 19th century Navier and Stokes formulated the equations that describe the flow of water and many other incompressible liquids under standard conditions. These equations now bearthe names of their inventors. A century later Leray developed the mathematical foundations of the modern theory of the Navier-Stokes equations both for planar and three-dimensional flows. He introduced the concept of generalized solution to the Cauchy problem and proved its existence for arbitrary (sufficiently regular) data and for an arbitrary time interval. This concept not only reflects the physical assumptions used when deriving the equations but it also forms the basis for the construction of powerful numerical methods.

Despite the undeniable success of the Navier-Stokes equations, there are many fluid-like incompressible materials that exhibit phenomena that can not be described by the Navier-Stokes equations. In order to describe these effects a number of models, which are more complicated than the Navier-Stokes equations, have been designed, developed, and used in relevant applications. The aim of this lecture is to survey recent developments, both in the area of theoretical continuum thermodynamics as well as in the field of PDE analysis, which have led to the development of Leray's programme beyond the Navier-Stokes equations.

PDE-CDT Hilary 2019 Schedule