The Kahler Ricci flow on complete non-compact Kahler manifolds and canonical Kahler metrics/structures

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Differential geometry is the study of space, its shape, and the interaction between the two.  My research focuses on two key areas of this study.  The first concerns the fundamental structure of space, and how its geometric shape ultimately affects the underlying fabric of the space itself.  The second is the problem of establishing the existence of certain ideal shapes.  These are geometric objects which have beautiful mathematical descriptions, but for which concrete examples are very hard to construct.  The main tool I use to study these problems in geometry is the Ricci flow equation.  This equation essentially prescribes a way to deform the shape of a given space into a geometrically nicer one.  The remarkable thing is that in many cases, the flow actually produces an ideal shape in this way!  The Ricci flow belongs to a class of equations known as geometric evolution equations, and the study of differential geomtery in this (and slightly more general) way is known as geometric analysis. The Ricci flow was first introduced by R.S. Hamilton in 1982, and has since been the focus of rapidly growing interest and efforts in the mathematical community.  This is especially due to the recent work of G. Perelman in the Ricci flow and its application to 3-dimensional geometry and topology. The Ricci flow also has fundamental applications to complex geometry. Despite this growing interest, there are still relatively few Canadian researchers in this exciting and fertile area. My research is a key component of the development of Ricci flow in both the mathematical community and in Canada.