Geometry of extremal Kahler metrics and geometric flows in Kahler settings

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In recent years, striking progress has been made in Kahler geometry. The famous Calabi conjecturestates that every Kahler manifold whose first Chern class C_1 has a definite sign will always have a Kahler Einstein metric with appropriate sign on its scalar curvature. This famous conjecture was proved by Yau in 1976 when C_1 =0, and independently by Yau and Aubin for the caseC_1 0. In Fano surface, the corresponding Calabi conjecture is solved by G. Tian in 1989. The high dimensional case is largely open till this day. This is an exciting area of mathematics in which we will see more important breakthroughs in the near future with impact both on the rest of mathematics and on physics. For instance, the work of Calabi-Yau directly provided mathematical foundation to mirror symmetry in the so called Calabi-Yau manifold. Therefore, we propose to study a network of problems centered around the existence of extremal Kahler metric, stability of the underlying polarized Kahler manifold, and other related areas. Extremal Kahler metrics is natural extension to the renown Calabi-Yau metrics.The principal investigator has been involved in the education and supervision of many graduate students at the University of Wisconsin and elsewhere. The proposer actively promotes the interaction of mathematicians from USA, Europe and Asian (particularly China, Japan and S.Korea). He has organized several international conferences in China where mathematicians from different background are invited. So this proposal will enhance our ability to train next generation of US based mathematicians.More broadly, The set of problems we propose here is key problems in differential geometry, which has strong impact to other fields of sciences like physics. According to Albert Einstein, the theory of gravity can be interpreted as the geometry of space-time. Thus the research in differential geometry is crucially important in physics and cosmology. The proposer's work, together with the works of other mathematicians and physicists, helps in understanding our Universe.