Geometry of extremal Kahler metrics and geometric flows in Kahler settings

近年来，卡勒几何取得了惊人的进展。著名的卡拉比猜想指出，每一个卡勒流形，其第一个陈氏类C_1有一个确定的符号，它的标量曲率上总是有一个带有适当符号的卡勒爱因斯坦度规。这个著名的猜想由Yau于1976年在C_1 =0时证明，并由Yau和Aubin分别在C_1 =0时证明。在Fano曲面上，对应的Calabi猜想由G. Tian（1989）求解。直到今天，高维的情况在很大程度上是开放的。这是一个令人兴奋的数学领域，在不久的将来，我们将看到更多重要的突破，这些突破将对其他数学和物理学产生影响。例如，Calabi-Yau的工作直接为所谓的Calabi-Yau流形中的镜像对称提供了数学基础。因此，我们建议以极值Kahler度量的存在性、底层极化Kahler流形的稳定性以及其他相关领域为中心，研究一个网络问题。极值Kahler度量是著名的Calabi-Yau度量的自然延伸。首席研究员参与了威斯康星大学和其他地方许多研究生的教育和监督。积极促进美国、欧洲和亚洲（特别是中国、日本和韩国）数学家的交流。他在中国组织了几次国际会议，邀请了不同背景的数学家。所以这个提议将提高我们培养下一代美国数学家的能力。更广泛地说，我们在这里提出的一组问题是微分几何中的关键问题，它对物理学等其他科学领域有很强的影响。根据爱因斯坦的理论，引力理论可以解释为时空的几何学。因此，微分几何的研究在物理学和宇宙学中具有极其重要的意义。这位提议者的工作，以及其他数学家和物理学家的工作，有助于理解我们的宇宙。

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.