Topology of symplectic four-manifolds

拓扑学是几何学的一个分支，如果两个几何对象可以从一个连续变形到另一个，则认为它们是相等的。 四维流形是没有任何尖角或边的四维几何对象。 它们具有根本的重要性，因为我们的宇宙本身可以用它们来建模。 四维流形的大多数基本例子是复曲面，它们由复数方程组定义。 现在我们知道绝大多数的四维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维流形，称为辛四维流形。 虽然大多数四维流形也不是辛的，但专家推测所有的四维流形都可以通过一些剪切和粘贴操作从辛四维流形建立起来。 辛四流形的完全分类仍然是我们目前的知识远远达不到的，但拓扑学家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.