The aim of the proposed project is to study a manifold with boundary, and some additional structure, in relation with the boundary values of the structure. This generic framework is a very natural approach to many important questions of differential geometry. Some typical cases are those of contact structures fillable by symplectic forms, complex geometry and Cauchy-Riemann boundary, Einstein geometry and conformal infinity. It is interesting to study how deformation theory of the boundary induces deformations of a filling. This problem is often related to some positive frequency condition, which is of particular interest for physicists. This will be used to study singular Yang-Mills connections of (a conjectural) Donaldson theory on Calabi-Yau 3-manifolds. More generally, we look at the picture of a moduli space on the manifold which projects on a corresponding moduli space defined on the boundary. Whenever it can be shown that the projection has a finite nonzero degree in some sense, we obtain a powerfull tool to prove the existence of fillings. Another way to tackle the questions of existence is to develop gluing theorems between moduli spaces. Some applications of gluing for Seiberg-Witten equations are already being developped thus giving new perspectives on contact geometry.
A physical theory is defined by a configuration space, together with objects (for example a metric) verifying particular equations. The state of a physical system is constrained by its configuration on the boundary of the space, or, at infinity. Therefore, it is a natural question to ask wether a theory is rich or empty: is it possible to find a physical system with a given behavior on the boundary, and if so, do we have many solutions? In other words is the system soft or rigid? Beyond the physical flavor, this approach, known as a field theory, has deep implications in mathematics. First, it is required to elaborate original tools in analysis, called moduli space theories, that have their own beauty. Secondly, the method relates very different problems. To illustrate that, we mention the following situation: we consider a 3-dimensional space (it could be the 3-dimensional sphere) bounding a 4-dimensional space (like the ball inside the sphere). Then, we look at knots on the boundary (they could be strands of DNA). These knots can be thought of as the boundary of a surface lying into the 4-dimensional space. Then, it can be shown that there is a subtle relation between the knot theory in dimension 3 and the theory of surfaces lying in a 4-dimensional space.
