Mathematical Sciences: 4-Manifolds and Symplectic Topology

这个项目的目的是更好地了解顺利 和辛四维流形 光滑四维流形是 局部同构于欧氏4-空间。 辛 流形被赋予了额外的结构， 面积和体积。 对于这两种类型的四维流形， 关于存在性和唯一性的基本问题， 这些问题正是本项目要解决的问题。 传统上，这两种类型的对象已被研究， 不同的方法，但首席研究员的 迄今为止的研究表明， approach. 事实上，他在这两个领域都解决了重大问题 通过简单的“剪切和粘贴”构造（对数变换和 沿沿着表面的连接和）。他打算应用这些方法， 相关技术以产生甚至更好的示例（例如， 单连通不可约辛4-流形 第二个贝蒂数字）。 这些和其他例子， 主要研究者已经产生，应该是可检测的 通过规范理论。 可能产生的其他例子 (such作为一个单连通的，辛的，非Kaehler 6-流形， 或辛4-流形，其分裂为以下的连通和： 不确定的碎片）可以通过基本手段检测到。 四维光滑流形是目前最小的 任何维度的流形。 仅在过去十年中 在了解可能的 这些物体的形状。 这是特别具有讽刺意味的是，我们的 我们自己的宇宙是一个四维流形的例子。 辛流形最近也受到了热烈的研究。 这些物体最初出现在与古典 力学，宏观物体的物理学，如机械 系统和轨道卫星。 随后，他们转向 在量子物理学中扮演着重要的角色，所以它们非常重要。 对那些试图理解 自然界的基本力量。 辛流形也是 纯数学家的兴趣，因为他们的出现在 诸如代数几何、规范理论和李 群论 首席研究员认为， 有很深的和很大程度上未被探索的联系之间的 辛流形和光滑4-流形的理论。 的 即时研究项目是利用这些联系， 阐明这两种理论。

The purpose of this project is better to understand smooth and symplectic 4-manifolds. Smooth 4-manifolds are spaces which are locally diffeomorphic to Euclidean 4-space. Symplectic manifolds are endowed with additional structure which keeps track of areas and volumes. For both types of 4-manifolds, the most basic questions of existence and uniqueness are still poorly understood; it is these questions which the project addresses. Traditionally, these two types of objects have been studied by rather different methods, but the principal investigator's research to date shows that much can be gained from a unified approach. In fact, he has solved major problems in both fields by simple "cut-and-paste" constructions (log transforms and connected sums along surfaces). He intends to apply these and related techniques to produce even better examples (for example, simply connected, irreducible symplectic 4-manifolds with small second betti numbers). These, and other examples which the principal investigator has already produced, should be detectable by means of gauge theory. Other examples which might be produced (such as a simply connected, symplectic, non-Kaehler 6-manifold, or a symplectic 4-manifold which splits as a connected sum of indefinite pieces) could be detected by elementary means. Smooth manifolds of dimension 4 are presently the least understood manifolds of any dimension. Only in the last decade has substantial progress been made in understanding the possible shapes of these objects. This is particularly ironic in that our own universe is an example of a 4-dimensional manifold. Symplectic manifolds have also come under intense study recently. These objects first appeared in connection with classical mechanics, the physics of macroscopic objects such as mechanical systems and orbiting satellites. Subsequently, they have turned out to play a deep role in quantum physics, so they are of great interest to physicists who are attempting to understand the fundamental forces of nature. Symplectic manifolds are also of interest to pure mathematicians because of their appearance in such diverse fields as algebraic geometry, gauge theory and Lie group theory. It is the principal investigator's belief that there are deep and largely unexplored connections between the theories of symplectic manifolds and smooth 4-manifolds. The instant research project is to exploit these connections to illuminate both theories.