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

辛、复流形、广义相对论和图的非线性分析

• 批准号: 1308244
• 负责人: Shing-Tung Yau
• 金额: $ 48.46万
• 依托单位: Harvard University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2013
• 资助国家: 美国
• 起止时间: 2013-07-01 至 2017-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1308244&HistoricalAwards=false
• 关键词: Nonlinear; Analysis; Sympletic; Complex; Manifolds

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.

Abstractaward：DMS 1308244，首席研究员：Shing-Tung Yauthere是我们在本提案中要执行的几个方向。 1。代数流形的符合性几何形状和时期：我们将广泛研究Tseng和我本人引入的新的共生几何学的新共生几何学，这可能被认为是bott-Chern共同体的概括。我们还将研究这种同种学的镜子对应物，并解释超对称方程在II A和II B理论中的作用。将计算许多示例以支持我们的理论。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歧管时期。将使用D模块的功率方法。 2。带有椭圆振动的Gromov-witten不变性和Calabi-yau歧管：我们将扩展我先前与Yamaguchi的工作，以研究字符串理论中更高环振幅的环。我们将为此环提供更精致的结构，并尝试构造其接近经典自身形式理论的类似特性。我们还喜欢以更深入的方式理解卡拉比河歧管的椭圆纤维结构变性的奇异结构。他们具有有趣的身体含义。 3。镜像的SYZ结构：我们喜欢研究这种结构中出现的仿射结构。有趣的方程式将被解决。 4。通过理解扭曲构建的平衡度量的结构：Nonkahler几何形状在弦理论中起着越来越重要的作用。我们将探索这种平衡的指标。 5。一般相对论中的准局部质量：Mu-Tao Wang，Po-ning Chen，我将继续研究Wang和我介绍的重要的准粒子质量。我们希望寻求这个群众的重要特性，并希望将其用于研究爱因斯坦方程的动态。 6。图理论的不变性：我们正在将内在的共同体学理论开发到定向图。我们认为，我们可以基于我们的构造来构建丰富的不变性。术语，我们正在应用几何方法来解决与物理学密切相关的新几何方面的重要问题，例如字符串理论和一般相对论。图理论的作品是一个新的方向，可以理解复杂网络，这些网络在计算机科学和其他数学领域中具有根本重要性。