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

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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.