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.

本文拟对光滑4流形的拓扑结构进行研究，将4流形中的一些尚未解决的问题分解为基本的易于理解的块，并运用复流形和辛流形理论的技术对这些块进行研究。他还计划研究校准流形和真正的代数变体。特别是，他计划研究某些类别的7维和8维流形（所谓的G2和Spin(7)流形）；通过研究其中的3维和4维子流形的某些族（即所谓的结合子流形和Cayley子流形），作者希望对低维流形的规范理论有一个整体的认识，并构建这些子流形的计数理论（类似于辛流形中全纯曲线的Gromov-Witten计数理论）。同时，本人希望继续从事实代数集的拓扑刻画课题。三维和四维流形，以及某些类型的七维和八维流形（G2和自旋(7)流形）是当前物理学家的兴趣，因为它们在理解时空和弦理论物理中起着核心作用。此外，代数集是描述方程中的拓扑空间的一种很好的方法，但并不是所有的拓扑空间都可以用这种方法来描述。建议计划描述所有可以被描述为实代数集的拓扑空间。