4-Manifolds, Calibrated Manifolds, Real Algebraic Varieties

4-流形、校准流形、实代数簇

• 批准号: 0505638
• 负责人: Selman Akbulut
• 金额: $ 10.8万
• 依托单位: Michigan State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2005
• 资助国家: 美国
• 起止时间: 2005-07-15 至 2008-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0505638&HistoricalAwards=false
• 关键词: Manifolds; Calibrated; Real; Algebraic; Varieties

Proposer plans to investigate the topology of smooth 4-manifolds by attacking some unsolved problems in 4-manifolds by decomposing them into basic easy to understand pieces (PALF's), and studying the pieces by applying techniques of complex and symplectic manifold theory. He also plans to work on calibrated manifolds, and on real algebraic varieties. In particular he plans to study on certain classes of 7 and 8 dimensional manifolds (so called G2 and Spin(7) manifolds); by studying the certain families of 3 and 4 dimensional submanifolds in them (so called associative and Cayley submanifolds) Proposer hopes to get a global understanding of the gauge theories of low dimensional manifolds, and construct a counting theory for these submanifolds (similar to Gromov-Witten counting theory of holomorphic curves in symplectic manifolds). Also, Proposer wants to continue to work on the project of topological characterization of real algebraic sets.Three and four dimensional manifolds, and certain classes of seven and eight dimensional manifolds (so called G2 and Spin(7) manifolds) are current interest of physicist because they play central role in understanding of space-time and the String theory physics. Also, algebraic sets are a nice way to describe topological spaces in equations, but not all the topological spaces can be described this way. Proposal plans to characterize all the topological spaces that can be described as real algebraic sets.

提议者计划通过将它们分解为基本易于理解的作品（Palf's），并通过应用复杂和符号歧管理论的技术来研究这些片段，从而通过攻击4个manifolds中的一些未解决的问题来研究光滑的4个manifolds的拓扑拓扑。他还计划研究经过校准的歧管和实际代数品种。他特别计划研究某些7和8维歧管的类别（所谓的G2和Spin（7）歧管）； by studying the certain families of 3 and 4 dimensional submanifolds in them (so called associative and Cayley submanifolds) Proposer hopes to get a global understanding of the gauge theories of low dimensional manifolds, and construct a counting theory for these submanifolds (similar to Gromov-Witten counting theory of holomorphic curves in symplectic manifolds).此外，提议者希望继续研究真实代数集的拓扑表征的项目。三个和四个维歧管，七个和八个维歧管的某些类别（所谓的G2和Spin（7）歧管）的某些类别是物理学家的当前兴趣，因为它们在理解时空和字符串理论物理学方面起着核心作用。同样，代数集是描述方程式拓扑空间的好方法，但并非所有拓扑空间都可以通过这种方式描述。建议计划表征所有可以描述为真实代数集的拓扑空间。