4-Manifolds, Calibrated Manifolds, Real Algebraic Varieties

Proposer计划研究光滑4-流形的拓扑结构，通过将4-流形中的一些未解决的问题分解为基本的易于理解的片段（PALF），并通过应用复流形和辛流形理论的技术来研究这些片段。他还计划工作的校准流形，并对真实的代数品种。特别是他计划研究某些类的7和8维流形（所谓的G2和Spin（7）流形）;通过研究其中的某些三维和四维子流形族，（所谓的结合和凯莱子流形）提案者希望得到一个全球性的理解规范理论的低维流形，并为这些子流形构造一个计数理论（类似于辛流形中全纯曲线的Gromov-Witten计数理论）。此外，Proposer希望继续致力于真实的代数集的拓扑特征的项目。三维和四维流形，以及某些类的七维和八维流形（所谓的G2和Spin（7）流形）是物理学家目前的兴趣，因为它们在理解时空和弦论物理中起着核心作用。此外，代数集是用方程描述拓扑空间的一种很好的方法，但不是所有的拓扑空间都可以用这种方法描述。建议计划的特点是所有的拓扑空间，可以描述为真实的代数集。

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.