4-Manifolds, Calibrated Manifolds, Real Algebraic Varieties

4-流形、校准流形、实代数簇

• 批准号: 0905917
• 负责人: Selman Akbulut
• 金额: $ 27.56万
• 依托单位: Michigan State University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-09-01 至 2015-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0905917&HistoricalAwards=false
• 关键词: Manifolds; Calibrated; Real; Algebraic; Varieties; Manifolds; Calibrated; Real; Algebraic; Varieties

The proposer plans to study topology of smooth 4- manifolds, calibrated manifolds, and real algebraic varieties. The plan to attack to unsolved problems in 4-manifold theory is to decompose 4-manifolds into basic, easy to understand, pieces, which are called Corks, Plugs and Palfs, and study these pieces by applying techniques from complex and symplectic manifold theory. Questions about knottedness and uniqueness of Corks are particularly important in this context. An immediate consequence of these techniques is the construction of exotic Stein manifolds. The P.I. also plans to work 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) the P.I. hopes to get a global understanding of the gauge theories of low dimensional manifolds, and to construct a counting theory for these submanifolds (similar to Gromov-Witten counting theory of holomorphic curves in symplectic manifolds). Also intended is to explain mirror duality in terms of G2 manifolds. Finally the P.I. wants to continue working on the project of topological characterization of real algebraic sets. 4-dimensional manifolds (spaces) appear to decompose into small basic pieces (particles) which we call "Corks" and "Plugs". These pieces determine the exotic structures of the underlying 4-manifold. One can think of corks and plugs as freely moving particles in 4-manifolds like Fermions and Bosons in physics, or little knobs on a wall to turn on and off; the ambient exotic lights in a room. The P.I. plans to study the structure of these Corks and Plugs. The P.I. also plans to study G2 and Spin(7) manifolds (certain 7 and 8 dimensional spaces), which are of current interest in physicists, because they play important role in the String theory and M-theory. The P.I. also plans to work on the problem of determining which topological spaces are algebraic sets. Making a topological space algebraic helps us to understand many of the properties of the space.

提议者计划研究光滑的4歧管，校准的歧管和实际代数品种的拓扑。在4个manifold理论中攻击未解决的问题的计划是将4个manifolds分解为基本，易于理解的碎片，这些碎片称为软木塞，塞子和palfs，并通过应用复杂和符号歧管理论的技术来研究这些部分。在这种情况下，有关打结和软木的独特性的问题尤其重要。这些技术的直接后果是构造异国风格的斯坦因歧管。 P.I.还计划在某些7和8维歧管的某些类别（所谓的G2和Spin（7）歧管）上工作。通过研究其中的某些3和4个维度子曼群的家族（所谓的Associative and Cayley Submanifolds）P.I.希望能够对低维流形的规格理论有一个全球的理解，并为这些子手机构建计数理论（类似于符合形态曲线的Gromov-witten计数理论）。还目的是用G2歧管来解释镜子二元性。最后是P.I. 希望继续研究真实代数集的拓扑表征。 4维歧管（空格）似乎分解为我们称为“软木”和“塞子”的小基本碎片（粒子）。这些碎片决定了基础4个manifold的外来结构。人们可以将软木和插头视为4个manifords中的颗粒，例如物理学中的福音和玻色子，或墙上的小旋钮打开和关闭；房间里的环境外来灯。 P.I.计划研究这些软木塞和插头的结构。 P.I.还计划研究G2和Spin（7）歧管（某些7和8维空间），这些歧管在物理学家中引起了人们的兴趣，因为它们在字符串理论和理论中都起着重要作用。 P.I.还计划处理确定哪些拓扑空间是代数集的问题。制作拓扑空间代数有助于我们了解该空间的许多特性。