The title is a wordplay on the way modern harmonic analysis can be understood. It is of course a classical approach in harmonic analysis to define boundaries and to find representations using boundary integrals. Thus we intend to develop the field of harmonic analysis itself. Second, harmonic analysis nourishes and is nourished by adjacent fields, among them are partial differential equations, functional analysis and geometry. Thus we are looking at the boundary of the field in its relations to other fields. The project will attack a number of problems in some of the most active themes of each of the fields.

The members include internationally renowned leaders in harmonic analysis, functional analysis and partial differential equations. The team contains also a number of active young mathematicians in those fields, geometry and functional calculus. It also includes a few non-permanent members (one post-doctorant and five PhD students).

The list of problems we plan to study range from questions in Harmonic Analysis (commutators, analysis on non-doubling spaces, Hardy spaces associated with operators, Riesz transforms, Multipliers and Bochner- Riesz means, elliptic boundary value problems wit minimal smoothness, Operators on tent spaces and applications to non-autonomous Cauchy problems), in Partial Differential Equations (extensions of Strichartz estimates and smoothing estimates to groups and manifolds in relation with curvature, and their applications to non-linear PDE's in this context, observability), in Functional analysis (Lifting of circle-valued maps, inequalities for Hodge systems, endpoint results for the inverting the divergence operator, Div-Curl Lemmas on manifolds, interpolation of Sobolev spaces on manifolds), in Geometry (sub-Riemannian geometry: curvature dimension , Harnack theory with minimal hypotheses, heat kernel and Riesz transforms of hypoelliptic operators; Hodge-de Rham operators in Riemannian geometry and applications). Methods of Harmonic Analysis will be central. Some of the problems are within reach before 2016, some are more exploratory.

This is why we ask for a 4 year project. It will allow smaller groups to gather and make progress on specific problems or on developing new sets of techniques and ideas. The project contains 4 milestones: one international exploratory conference in its beginning year, 2 small meetings of the members of the project in the second and fourth years to discuss advances and a scientific synthesis.