Symplectic Cobordism Relations on Contact Manifolds

微分几何是研究“光滑形状”，例如没有粗糙边缘或尖锐弯曲的曲面。表面是二维物体，人们可以类似地想象一维的光滑形状，如直线、曲线或圆。更难想象的是，但可以用精确的数学术语描述，是任意维数的光滑形状：这些物体被称为“流形”。二维流形的一个具体例子是圆盘，即圆内的区域，其“边界”是一维流形，即圆。类似地，对于任何正整数n，n维流形可以具有（n-1）维流形的边界。我们可以很容易地画出的所有三维流形都是这种类型的：例如，如果我们想象三维空间中的任何表面，例如球体或“环面”。（甜甜圈表面的形状），那么该表面内的区域是一个三维流形，其边界是表面。我们现在可以问一个关于流形的最基本的问题：给定一个n维流形，它是某个物体的边界吗？这实际上不仅仅是一个几何问题，而是一个“拓扑”问题，这是一种研究几何物体“整体形状”的方法。正如上面给出的例子，我们可以很容易地想象到的大多数二维流形都是它们所包围的三维区域的边界。但举一个更有趣的例子，我们可以试着想象一个“克莱因瓶”：这是一个表面，它是由一个普通的瓶子和弯曲它的开口周围，并通过玻璃进入内部，然后通过向上弯曲底部将开口连接到瓶子的底部。结果是一个表面不是任何东西的边界，因为它的内部与外部没有区别;就像莫比乌斯带，但本身封闭。该提案的主题涉及上述关于边界问题的更详细版本：我们处理偶数维的特殊类型的流形，称为“辛”流形，它们的奇维边界称为“接触”流形。辛流形的概念最初来自物理学：一个世纪前，辛流形被认为是研究汉密尔顿在世纪对牛顿经典力学的重新表述的自然几何背景。今天，辛流形被认为是有趣的，他们自己的权利，他们保留了连接到物理学，但一个非常不同的和非经典的排序：通过研究具有切触边界的辛流形中的某些特殊曲面，可以定义一个所谓的“辛场论”。（或简称“SFT”），它与现代物理学用来描述基本粒子及其相互作用的一些理论有着强烈但神秘的相似之处。与这些理论不同，SFT不能帮助我们预测粒子加速器中会发生什么，但它可以帮助我们回答“辛和接触拓扑”领域的一个基本问题：给定一个接触流形，它是任何辛流形的边界吗？更一般地说，研究接触流形本身的一种方法是考虑以下关系：我们说两个这样的流形是“辛共边的”，如果它们形成辛流形边界的两个独立部分。两个给定的切触流形是否共边的问题帮助我们理解什么样的切触流形可以首先存在，辛场论是我们研究这个问题的最强大的方法之一。因此，这个项目的目标是使用这个和相关的工具来学习尽可能多的关于接触流形上的辛配边关系。由于之前关于该主题的大多数结果都集中在具有3维边界的4维流形上，因此我们的目标特别是在更高维度上获得新的见解。

Differential geometry is the study of "smooth shapes", e.g. curved surfaces that have no rough edges or sharp bends. A surface is a 2-dimensional object, and one can similarly imagine smooth shapes that are 1-dimensional, such as a line, or curve, or circle. What is much harder to imagine, but can nonetheless be described in precise mathematical terms, is a smooth shape in an arbitrary number of dimensions: these objects are called "manifolds".A specific example of a 2-dimensional manifold is a disk, i.e. the region inside a circle, and its "boundary" is a 1-dimensional manifold, namely the circle. Similarly, for any positive integer n, an n-dimensional manifold may have a boundary which is an (n-1)-dimensional manifold. All the 3-dimensional manifolds that we can easily picture are of this type: e.g. if we imagine any surface in 3-dimensional space, such as a sphere or a "torus" (the shape of the surface of a doughnut), then the region inside that surface is a 3-dimensional manifold whose boundary is the surface.We can now ask one of the most basic questions concerning manifolds: given an n-dimensional manifold, is it the boundary of something? This is actually not just a geometric question, but really a question of "topology", which is a certain way of studying the "overall shape" of geometric objects. As in the example given above, most 2-dimensional manifolds that we can easily imagine are boundaries of the 3-dimensional regions they enclose. But for a more interesting example, we can try to imagine a "Klein bottle": this is a surface formed by taking an ordinary bottle and bending its opening around and through the glass into the inside, then connecting the opening to the floor of the bottle by curving the floor upward. The result is a surface that is not a boundary of anything, as its inside is not distinct from its outside; like a Moebius strip, but closed in on itself.The subject of this proposal concerns a more elaborate version of the above question about boundaries: we deal with a particular type of manifold in an even number of dimensions, called "symplectic" manifolds, and their odd-dimensional boundaries are called "contact" manifolds. The idea of a symplectic manifold comes originally from physics: a century ago, symplectic manifolds were understood to be the natural geometric setting in which to study Hamilton's 19th century reformulation of Newton's classical mechanics. Today symplectic manifolds are considered interesting in their own right, and they retain a connection to physics, but of a very different and non-classical sort: by studying certain special surfaces in symplectic manifolds with contact boundary, one can define a so-called "Symplectic Field Theory" (or "SFT" for short), which bears a strong but mysterious resemblance to some of the theories that modern physics uses to describe elementary particles and their interactions. Unlike those theories, SFT does not help us to predict what will happen in a particle accelerator, but it can help us answer a basic question in the area of "Symplectic and Contact Topology": given a contact manifold, is it the boundary of any symplectic manifold?More generally, one way to study contact manifolds themselves is to consider the following relation: we say that two such manifolds are "symplectically cobordant" if they form two separate pieces of the boundary of a symplectic manifold. The question of whether two given contact manifolds are cobordant helps us understand what kinds of contact manifolds can exist in the first place, and Symplectic Field Theory is one of the most powerful methods we have for studying this. The goal of this project is thus to use this and related tools to learn as much as we can about the symplectic cobordism relation on contact manifolds. Since most previous results on this subject have focused on 4-dimensional manifolds with 3-dimensional boundaries, we aim especially to gain new insights in higher dimensions.