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

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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.