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流形连续变形到另一个几何对象，则它们被认为是相等的。四流形是没有任何尖角或边缘的四维几何对象。它们具有基本的重要性，因为我们的宇宙本身可以用它们来建模。四流形最基本的例子是复杂曲面，它由复数方程组定义。现在我们知道，绝大多数四流形不是复杂曲面。我的研究将研究一类更一般的四流形，所谓辛四流形。虽然大多数四流形也不是辛的，但专家们猜测，所有的四流形都可以通过一些剪切粘贴操作由辛四流形建立起来。我们目前的知识还远远达不到辛四流形的完整分类，但拓扑学家近年来取得了重大进展。经典纽结理论处理三维空间中有节的一维曲线的分类。我提出的研究将研究辛四维流形中的有节的二维曲面。他注意到纽结对象和包含它的周围空间的维度都增加了一，因此，这类纽结的二维曲面的分类是一项相当艰巨的任务。我的研究将集中在从包含它们的环境辛四维流形继承辛子结构的二维曲面子类。专家们猜想，这个子类中的纽结二维曲面的数量和种类衡量了包含这些纽结曲面的环境辛四维流形的拓扑复杂性。我以前的研究已经证明了这一猜想的一些证据，我希望我未来在这个方向的工作可以为辛四维流形的分类做出贡献。

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.    Classical knot theory deals with classifying knotted one-dimensional curves in three-dimensional space.  My proposed research will study knotted two-dimensional surfaces in symplectic four-manifolds.  Note that dimensions of both the knotted object and the ambient space containing it have gone up by one, and consequently the classification of such knotted two-dimensional surfaces turns out to be a rather daunting task.  My research will focus on the subclass of two-dimensional surfaces that inherit a symplectic substructure from the ambient symplectic four-manifold containing them.  Experts conjecture that the number and the variety of knotted two-dimensional surfaces within this subclass measure the topological complexity of the ambient symplectic four-manifold containing those knotted surfaces.  My previous research has shown some evidence of this conjecture and I hope that my future work in this direction can contribute towards the classification of symplectic four-manifolds.