Mathematical Sciences: Analytic Aspects of Pseudo-Holomorphic Curves

点击翻译按钮获取中文摘要

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.