Geometric analysis on non-compact and singular spaces

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Geometric analysis is about the study of partial differential equations in geometric settings. On one hand, one can study differential equations like the wave equation or the heat equation on curved spaces and look how the geometry and the topology influence the solutions to these equations. On the other hand, geometric analysis also plays an important role in the solutions of certain geometric problems, typically the search on a given space of a Riemannian metric (a geometry) with nice properties like a uniform curvature. Indeed, quite often, one obtains such metrics by solving a partial differential equation, for instance one can think of Einstein's equation in the theory of general relativity. Since singularities are omnipresent in science and every day life, from breaking waves to the notion of black holes in general relativity, it is very natural to consider more generally curved spaces admitting singularities. This will be the focus of the proposed research program. Namely, we will study various partial differential equations, some coming from physics, some from geometric problems, in situations where the underlying curved space has singularities or infinite ends. Even for simple singularities like the tip of a cone or an edge, this can bring many serious difficulties, but also interesting new phenomena. A good understanding of the influence of singularities on the solutions to certain partial differential equations can in turn be useful to detect the presence of singularities, for instance in medical imaging. We will explore three directions of research. In one direction, we will study the behavior of solutions to geometric equations near a singularity, which at the same occasion shall give a better geometric description of that singularity. In another direction, we will try to 'hear' the presence of a singularity by studying the wave equation, the spectrum of the Laplacian, and more generally invariants built out of this spectrum. Finally, through geometric equations, we will study how the presence of singularities influence the topology of the underlying space and vice versa.