Nonlinear Analysis on Sympletic, Complex Manifolds, General Relativity, and Graphs

摘要奖号：DMS 1308244，主要研究者：姚成东在这个建议中，我们将进行几个方向。1.辛几何和代数流形的周期：我们将广泛地研究Tseng和我引入的辛几何的新的上同调不变量，它可以被认为是Bott-Chern上同调的推广。我们还将研究这个上同调的镜像对应物，并解释在这个新的上同调中，Ⅱ A和Ⅱ B理论中的超对称方程的作用。将计算出许多例子来支持我们的理论。为了更深入地理解镜像对称理论，我们还将计算非复曲面的代数流形的周期。将使用D模块的幂方法。2. Gromov-Witten不变量和Calabi-Yau流形与椭圆振动：我们将扩展我以前的工作与山口研究环的更高的循环振幅弦理论。我们将给这个环更精细的结构，并试图构建类似的性质，它是接近经典的自守形式理论。我们还想更深入地理解Calabi-Yau流形椭圆纤维结构退化的奇异结构。它们具有有趣的物理意义。3.镜像流形的SYZ构造：我们喜欢研究这种构造中出现的仿射结构。有趣的方程将被求解。4.通过理解扭量构造来构造平衡度规：Nonkahler几何在弦理论中扮演着越来越重要的角色。我们将探索这种平衡的指标。5.广义相对论中的准定域质量：王慕涛、陈柏宁和我将继续研究我和王慕涛介绍的重要准定域质量。我们希望能找到这个质量的重要性质，并希望利用它来研究爱因斯坦方程的动力学。6.图论的不变量：我们正在发展有向图的内在上同调理论。我们相信我们可以在我们的基础上构造丰富的不变量。总体而言，我们正在应用几何方法来解决与物理学密切相关的几何新前沿的重要问题，例如弦理论和广义相对论。图论的工作开辟了理解复杂网络的新方向，复杂网络在计算机科学和其他数学领域具有根本的重要性。

AbstractAward: DMS 1308244, Principal Investigator: Shing-Tung YauThere are several directions that we shall carry out in this proposal. 1. Symplectic geometry and periods of algebraic manifolds: We shall study extensively the new cohomological invariants for symplectic geometry introduced by Tseng and myself, which may be considered as generalization of Bott-Chern cohomology. We shall also study the mirror counterpart of this cohomology and interpret the role of the supersymmetric equations in II A and II B theories in this new cohomology. Many examples will be calculated to support our theory. For deeper understanding of the theory of mirror symmetry, we shall also calculate periods of algebraic manifolds which are not toric. Power method of D-modules will be used. 2. Gromov-Witten invariants and Calabi-Yau manifolds with elliptic vibrations: We shall extend my previous work with Yamaguchi to study the ring of higher loop amplitude in string theory. We shall give more refined structure to this ring and try to construct analogous properties of it that are close to classical automorphic form theory. We also like to understand in a deeper manner the singular structure of the degeneration of elliptic fiber structure of Calabi-Yau manifolds. They have interesting physical meaning. 3. SYZ construction of mirror manifolds: We like to study the affine structure that appears in this construction. Interesting equations will be solved. 4. Construction of balanced metric through understanding of twistor construction: Nonkahler geometry is playing an increasingly important role in string theory. We shall explore such balanced metrics. 5. Quasilocal mass in general relativity: Mu-Tao Wang, Po-Ning Chen, and I will continue to study the important quasilocal mass that Wang and I introduced. We hope to seek the important property of this mass and hopefully use it to study dynamics of Einstein equation. 6. Invariants of graph theory: We are developing intrinsic cohomology theory to directed graphs. We believe that we can construct rich invariant based on our construction.Overall, we are applying geometric methods to solve important questions in new fronts of geometry that are closely related to physics, such as string theory and general relativity. The works on graph theory pioneer a new direction to understand complex networks which have fundamental importance in computer science and other areas of mathematics.