Topology of symplectic four-manifolds

拓扑学是几何学的一个分支，如果两个几何物体可以连续地从一个变形到另一个，则认为它们是相等的。四维流形是没有任何尖角或边缘的四维几何对象。它们是至关重要的，因为我们的宇宙本身可以用它们来建模。四流形的最基本的例子是复曲面，它是由复数方程系统定义的。现在我们知道绝大多数的四流形都不是复杂曲面。我提议的研究将研究一类更一般的四流形，称为辛四流形。虽然大多数四流形也不是辛流形，但专家们推测，所有的四流形都可以通过一些剪切和粘贴操作从辛四流形建立起来。辛四流形的完整分类仍然远远超出我们目前的知识范围，但拓扑学者近年来已经取得了重大进展。

Topology is a branch of geometry where two geometric objects are considered to be equal if they can be continuously deformed from one to the other. Four-manifolds are four-dimensional geometric objects that do not have any sharp corner or an edge. They are of fundamental importance as our universe itself can be modeled using them. Most elementary examples of four-manifolds are the complex surfaces, which are defined by systems of equations in complex numbers. It is now known that the vast majority of four-manifolds are not complex surfaces. My proposed research will study a more general class of four-manifolds, called symplectic four-manifolds. Although most four-manifolds are not symplectic either, experts conjecture that all four-manifolds can be built up from symplectic four-manifolds via some cut-and-paste operations. The complete classification of symplectic four-manifolds is still far out of our reach with our current knowledge, but topologists have been making significant progress in recent years.