Mathematical Sciences: Analytic Aspects of Pseudo-Holomorphic Curves

这个提案包含了三个关于伪全纯曲线理论解析方面的项目。第一种是计算代数几何枚举不变量的解析方法；第二个项目旨在确定气泡树可以作为j全纯映射的极限的新约束；第三个课题是关于辛四流形的taubs型不变量。特别地，研究者对获得比Seiberg-Witten不变量更精细的辛四流形的不变量感兴趣。根据爱因斯坦的广义相对论，我们的宇宙可以被认为是一个四维的弯曲空间。因此，四流形是这个四维弯曲空间的数学抽象和推广。它们是通用的，因为不同的曲率和形状属性是可能的；它们是抽象的，因为四流形不一定位于环境空间中。近年来，数学家们一直试图根据各种方案对四流形进行分类；提出的研究着眼于这些流形的所谓可微结构。

9626245 Parker This proposal contains three projects on the analytic aspects of the theory of pseudo-holomorphic curves. The first is an analytic approach to computing the enumerative invariants of algebraic geometry; the second project aims to identify new constraints on which bubble trees can arise as limits of J-holomorphic maps; the third project has to do with Taubes-type invariants of symplectic four-manifolds. In particular, the investigator is interested in obtaining invariants of symplectic four-manifolds finer than the Seiberg-Witten invariants. Our universe, according to Einstein's general theory of relativity, can be thought of as a four-dimensional curved space. Four-manifolds then are a mathematical abstraction and generalization of this four dimensional curved space. They are general in that different curvature and shape properties are possible; they are abstract in that a four manifold does not necessarily sit inside an ambient space. Mathematicians have been trying to classify four-manifolds according to various schemes in recent years; the proposed research looks at so called the differentiable structure of these manifolds.