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

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.

提出者计划研究光滑4-流形、校准流形和真实的代数簇的拓扑。解决四维流形理论中尚未解决的问题的计划是将四维流形分解成基本的、易于理解的块，称为Corks、Plug和Palfs，并应用复流形和辛流形理论的技术来研究这些块。在这种情况下，关于软木塞的打结和独特性的问题尤为重要。这些技术的直接后果是建设异国情调的斯坦流形。私家侦探也计划工作的某些类的7和8维流形（所谓的G2和自旋（7）流形）。通过研究其中的某些3维和4维子流形族（所谓结合子流形和Cayley子流形），得到了P.希望能对低维流形的规范理论有一个全面的了解，并为这些子流形建立一个计数理论（类似于辛流形中全纯曲线的Gromov-Witten计数理论）。还打算解释镜像对偶的G2流形。最后，PI 希望继续致力于真实的代数集的拓扑表征项目。4-维流形（空间）似乎分解成小的基本块（粒子），我们称之为“软木塞”和“塞子”。这些部分决定了底层4-流形的奇异结构。人们可以把软木塞和塞子想象成在四维流形中自由移动的粒子，就像物理学中的费米子和玻色子，或者墙上的小旋钮来打开和关闭;房间里的环境异国情调的灯光。私家侦探计划研究这些软木塞的结构。私家侦探还计划研究G2和Spin（7）流形（某些7维和8维空间），这是物理学家目前感兴趣的，因为它们在弦理论和M理论中起着重要作用。私家侦探还计划工作的问题，确定哪些拓扑空间是代数集。使拓扑空间代数化有助于我们理解空间的许多性质。