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

微分几何是研究空间，它的形状，以及两者之间的相互作用。我的研究集中在这项研究的两个关键领域。第一个关注空间的基本结构，以及它的几何形状最终如何影响空间本身的底层结构。第二个问题是确定某些理想形状存在的问题。这些几何物体有漂亮的数学描述，但具体的例子很难构建。我用来研究这些几何问题的主要工具是里奇流动方程。这个方程本质上规定了一种将给定空间的形状变形成几何上更好的形状的方法。值得注意的是，在许多情况下，流实际上以这种方式产生理想的形状！里奇流属于一类被称为几何演化方程的方程，以这种（稍微更一般的）方式研究微分几何被称为几何分析。Ricci流最早是由R.S. Hamilton在1982年提出的，从那以后，它就成为数学界迅速增长的兴趣和努力的焦点。特别是由于G. Perelman最近在Ricci流方面的工作及其在三维几何和拓扑中的应用。里奇流在复杂几何中也有基本的应用。尽管人们的兴趣越来越浓厚，但在这个令人兴奋和肥沃的领域，加拿大的研究人员仍然相对较少。我的研究是利玛窦流在数学界和加拿大发展的关键组成部分。

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.