4-Manifolds, Calibrated Manifolds, Real Algebraic Varieties

本文拟研究光滑4-流形、标定流形和实代数变体的拓扑结构。解决四流形理论中尚未解决的问题的计划是将四流形分解为基本的，易于理解的块，称为Corks， plug和Palfs，并通过应用复流形和辛流形理论中的技术来研究这些块。关于软木塞的结度和独特性的问题在这种情况下尤为重要。这些技术的直接结果是构造奇异的斯坦流形。P.I.也计划研究某些7维和8维流形（所谓的G2和自旋(7)流形）。通过研究它们中的某些3维和4维子流形族（即所谓的结合子流形和Cayley子流形），P.I.希望对低维流形的规范理论有一个全面的认识，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）。还打算用G2流形来解释镜像对偶性。最后，P.I.想继续研究实代数集的拓扑表征。四维流形（空间）似乎分解成小的基本块（粒子），我们称之为“软木塞”和“塞”。这些部件决定了底层4歧管的奇特结构。人们可以把软木塞和塞子想象成四流形中自由运动的粒子，就像物理学中的费米子和玻色子，或者墙上用来开关的小旋钮；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塞子的结构。P.I.还计划研究G2和自旋(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。pi还计划研究确定哪些拓扑空间是代数集的问题。将拓扑空间化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.