On the Geometry of Calabi-Yau Moduli and Kahler-Einstein manifolds

获奖：DMS 1206748，首席研究员：陆志勤。本课题的目标是继续研究微分几何及相关领域的几个基本问题。长期目标是研究高阶BCOV猜想以及Kahler-Einstein度量的存在性与复流形稳定性之间的关系。短期目标之一是研究稳定性的不同符号。特别地，作者将研究Donaldson-Futaki不变量与Tian-Futaki不变量之间的关系。提议者认为，像Futaki不变量一样，这两个不变量都是组合的。他还将推广Tian-Yau-Zelditch展开，研究间隙问题，研究量子层的谱问题，并研究与平均曲率流相关的一些基本线性代数问题。所提出的问题在微分几何和数学物理中是重要的。它对我们对宇宙的理解产生了重大影响。和以前一样，他将通过演讲和建议学生，向学生和公众传达他的数学见解。他将继续鼓励更多的人，尤其是女性和少数民族，学习数学。为了实现这一目标，申请人参与了许多教育活动，如加州数学计数，UCI食蚁兽数学俱乐部，UCI物理科学学院导师计划等。他参与了本科和研究生课程的创新。

AbstractAward: DMS 1206748, Principal Investigator: Zhiqin LuThe object of this proposal is to continue to study several fundamental problems in differential geometry and related fields. The long-term goal is to study the higher order BCOV conjecture and relations between the existence of Kahler-Einstein metrics and the stability of complex manifolds. One of the short-term goals is to study the different notations of stability. In particular, the proposer will study the relations between the Donaldson-Futaki invariant and the Tian-Futaki invariant. The proposer believes that, like the Futaki invariant, both invariants are combinatorial. He will also generalize the Tian-Yau-Zelditch expansion, study the gap problem, study the spectral problem of quantum layers, and study some fundamental linear algebra problems related to the mean curvature flow.The proposed problems are important in differential geometry and mathematical physics. It has significant impacts in our understanding of the Universe. As before, he will convey his mathematical insights to students and to the general public by giving talks and advising students. He will continue to encourage more people, especially women and minorities, to study mathematics. To achieve the goal, the proposer is involved in many educational activities such as the California Math Counts, the UCI Anteater Math Club, the UCI School of Physical Sciences Mentor Program, etc. He is involved in the innovation of both undergraduate and graduate courses.