Finding canonical metrics in complex differential geometry

This proposal is in the area of complex differential geometry, a prominent field of mathematics. It stands at the intersection of differential and algebraic geometry.

The basic objects are manifolds, spaces that locally look like flat space, and vector bundles over them - a collection of vector spaces parametrised by a manifold. In complex differential geometry one seeks optimal notions of distance, so-called canonical metrics. In higher dimensions, canonical metrics may or may not exist.

The key question is to determine whether or not a given space has a canonical metric, a very challenging problem.

The Yau-Tian-Donaldson conjecture stands at the heart of this problem, and relates the existence of a solution to algebro-geometric notions of stability.The aim of this research proposal is to give several new constructions of canonical metrics for complex manifolds, holomorphic vector bundles and families of such objects.

It also seeks to show connections of the existence of these metrics, a solution to a PDE, with purely algebraic notions, for an equation for families of canonical metrics.

This will be approached mainly with techniques from perturbative and variational PDE theory and algebraic geometry, but will also use some computational methods and probability theory.

The proposal seeks to develop new techniques for well studied equations, and to apply more well known techniques to new equations, to advance the constructions and the theory of canonical metrics in a major way.The action would give a unique opportunity for a reciprocal transfer of knowledge as part of a prominent research group in the field, whose research focus and strengths differ from that of the ER.

It would provide the ER with the independence needed to form his own research group in the future, and expand the ER's academic network through new connections. Though currently working in Europe, the ER was previously in North America. The fellowship would allow the ER to remain within the EU.