Geometric analysis via manifolds with corners

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From breaking waves to the notion of black holes in Einstein's theory of relativity, singularities are omnipresent in science and every day life. Mathematics, more specifically geometry and analysis, provide the right language to describe and study them, especially the way they affect solutions to partial differential equations (PDE's), for instance the heat equation or the wave equation. In this proposal, we intend to use the notion of manifolds with corners to study this sort of questions. Indeed, manifolds with corners can often be used to resolve the singularities of a space by adding suitable boundary hypersurfaces. When one wants to solve a PDE, these boundary hypersurfaces become very useful to give asymptotic models of the PDE. To solve the PDE, a natural strategy is then to first solve the model problems at each boundary hypersurfaces, then patch them together to obtain a good approximate solution of the PDE, and finally solve exactly the PDE using functional analytical methods. Using this approach, this proposal will focus in particular on developing analytical tools to study the space of magnetic monopoles. This is a subtle space, since monopoles can sometime be seen as distinct particles, but typically can coalesce in one indivisible object. Nevertheless, there are strong evidences that there is a manifold with corners describing this space and the way monopoles scatter at infinity. One important goal of the proposal is to construct a suitable calculus of operators for such a space that will allow to solve natural geometric PDE's on it. A good understanding of these solutions will allow to verify predictions coming from theoretical physics and based on the principle of S-duality in string theory. This calculus of operators will also be useful to study other configuration spaces. At the same time, it will provide new examples of Calabi-Yau spaces, which are spaces with vanishing Ricci curvature, meaning that they are solutions of the Riemannian analog of Einstein's equation in general relativity. In another direction, still using manifolds with corners, the proposal will study spectral invariants in the presence of singularities. Like echolocation for bats or dolphins, the spectrum of a geometric operators and spectral invariants built out of it allow in many instances to `hear' the shape of space. One important example of spectral invariant is analytic torsion, which can detect information about the topology of the space. The proposal will study in particular how analytic torsion is affected by the presence of edge singularities. Motivated by hyperbolic geometry and number theory, it will also study analytic torsion on spaces with cusps singularities and try to determine what topological information it can recover from the space. The methods developed will also be useful to study other spectral invariants, like the eta invariant or the regularized determinant of elliptic operators.