University of Warwick

Title: An introduction to rectifiability in metric spaces

The main objective is to give a concise introduction to the theory of rectifiability in an arbitrary metric space and to draw comparisons to classical rectifiability in Euclidean space. We will begin with Kirchheim's description of rectifiable subsets of a metric space. We will then discuss sufficient conditions for rectifiability: bi-Lipschitz decompositions of functions; rectifiability from topology; Alberti representations. Finally we will characterise rectifiable subsets of a metric space in terms of flat tangent spaces.

Smith College

Title: An introduction to nonlocal PDE in metric measure spaces

In a seminal 2007 paper, Caffarelli and Silvestre  explored a way to study  (nonlocal) fractional powers of the Euclidean Laplacian,  by means of the Dirichlet to Neumann map for a suitable class of degenerate elliptic (local) PDE in the upper half-space. 

In these lectures we will explore a way to extend this approach to the non-smooth setting and study nonlocal PDE in doubling metric measure spaces. The main focus will be on a review of the background concepts involved in this extension: Gromov hyperbolic spaces and their visual boundaries, hyperbolic fillings, uniformization theorems, first order calculus in metric measure spaces, and the associated Dirichlet and Neumann problems.

University of Bath

Title: Title: Harmonic analysis on nilpotent Lie groups

Harmonic and global analysis on nilpotent Lie groups and on sub-Riemannian manifolds are topics which are very much intertwined for scientific and historical reasons. We will discuss various aspects of this analysis, focussing on pseudo-differential theories with applications to spectral properties. 

University of Tromsø

Title: Cartan geometry and Tanaka theory

 Cartan geometry is a setup to cover a varieity of geometric structures, like Riemannian, conformal, projective, etc. Tanaka theory expands this to nonholonomic distributions, including finite type distributions, sub-Riemannian metrics and CR geometries. Both theories are curvature deformations of the homogeneous spaces, and include as a particular case parabolic geometries. This yields important tools for analysis of automorphisms. I will discuss solutions of the equivalence problem, which lead to description to curvatures/torsions, as well as maximally and submaximally symmetric models.