4-Manifolds, Calibrated Manifolds, Real Algebraic Varieties

作者计划研究光滑4-流形、刻度流形和实代数簇的拓扑。解决4-流形理论中尚未解决的问题的计划是将4-流形分解成基本的、易于理解的片段，这些片段被称为软木、塞和PARF，并应用复流形理论和辛流形理论的技巧来研究这些片段。在这种情况下，关于软木塞的打结性和独特性的问题尤其重要。这些技术的一个直接后果就是构建了充满异国情调的斯坦因流形。P.I.还计划研究某些类别的7维和8维流形(即所谓的G2和Spin(7)流形)。通过研究其中的三维子流形和四维子流形(所谓的结合子流形和Cayle子流形)，P.I.希望对低维流形的规范理论有一个全面的理解，并构造一个关于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)。也是为了用G2流形来解释镜像对偶。最后，P.I.希望继续研究实代数集的拓扑刻画。4维流形(空间)似乎分解成小的基本碎片(粒子)，我们称之为“软木”和“塞子”。这些碎片决定了潜在的4-流形的奇异结构。人们可以把软木塞和插头想象成四维空间中自由运动的粒子，就像物理学中的费米子和玻色子，或者墙上打开和关闭的小旋钮；房间里充满异国情调的灯光。私家侦探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)流形(某些7维和8维空间)，这是目前物理学家感兴趣的，因为它们在弦理论和M理论中发挥着重要作用。P.I.还计划研究确定哪些拓扑空间是代数集的问题。使拓扑空间成为代数有助于我们理解空间的许多性质。

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.