Ricci flow of manifolds with singularities at infinity

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This proposal concerns geometric flows, which is a subject that lies at the interface of differential geometry, analysis, topology and the theory of nonlinear partial differential equations (PDEs). More specifically, we will consider Ricci flow, which is a way of taking a curved space, known as a Riemannian manifold, and deforming it in time to make it more uniform.The importance of the field cannot be overstated. Ricci flow is famous for solving a string of major problems such as the 100 year old Poincaré conjecture, which had a $1,000,000 bounty attached to it, and Thurston's geometrisation conjecture, but the potential extent of its applications lies far beyond. Up until now, the theory has focussed almost exclusively on manifolds that are compact, or that have artificial constraints on their behaviour at infinity such as a uniform upper curvature bound or a positive uniform lower bound on the volume of every unit ball. This proposal is directed towards the next wave of applications. To realise these we must understand flows that are singular at infinity, and to do this we will need to advance the theory of nonlinear PDEs and understand better their interaction with geometry. We will require a collection of innovations, including new curvature estimates and a better understanding of the geometry at infinity of positively curved manifolds.Even partial success along these lines will transform the applicability of the field. Progress will give us an understanding of the geometry and topology of open manifolds without artificial asymptotic constraints on their geometry. We give some illustrative examples of major open problems that would fall to the advances that we envisage, such as Yau's Uniformisation Conjecture, and describe a route to achieve them.The proposal has some highly ambitious objectives. However, it also contains a collection of conjectures and problems, of varying difficulty, that push on many fronts against the central aim of understanding flows with unbounded curvature, and collapsing behaviour, at infinity. What is particularly exciting about this research direction is that only in the past few years have we been successful in developing the foundational theory to make this feasible. Thanks to the work of several international teams, including that of the PI and M. Simon in their resolution of the Anderson-Cheeger-Colding-Tian conjecture in 3D, we now have a clear idea of the required a priori estimates, which differ substantially from the scale-invariant estimates proved thus far, and we finally have a roadmap towards establishing them.