Symplectic Cobordism Relations on Contact Manifolds

微分几何是研究“光滑形状”的学科，例如，没有粗糙边缘或尖锐弯曲的曲面。曲面是二维对象，人们可以类似地想象一维的平滑形状，如直线、曲线或圆。很难想象，但仍然可以用精确的数学术语来描述，在任意数量的维度上的光滑形状：这些对象被称为“流形”。2维流形的一个具体例子是圆盘，即圆内的区域，而它的“边界”是1维流形，即圆。类似地，对于任意正整数n，n维流形的边界可以是(n-1)维流形。我们可以很容易地描绘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(甜甜圈的表面的形状)，那么该表面内的区域就是其边界是表面的三维流形。现在我们可以问关于流形的最基本的问题之一：给定一个n维流形，它是某物的边界吗？这其实不仅仅是一个几何问题，而是一个真正的“拓扑”问题，这是研究几何物体“整体形状”的一种方式。正如上面给出的例子，我们可以很容易地想象大多数2维流形是它们所包围的3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”：这是一个普通瓶子的表面，它的开口绕着玻璃弯曲，穿过玻璃进入瓶子内部，然后通过向上弯曲地板将开口连接到瓶底。结果是一个不是任何东西的边界的表面，因为它的内部和外部没有区别；就像Moebius带，但它本身是封闭的。这个提议的主题涉及上面关于边界问题的一个更精细的版本：我们处理偶数维的特定类型的流形，称为“辛”流形，而它们的奇数维边界称为“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19世纪对牛顿经典力学的重新表述的自然几何背景。今天，辛流形本身就被认为是有趣的，它们与物理学保持着一种非常不同的、非经典的联系：通过研究具有接触边界的辛流形中的某些特殊表面，人们可以定义所谓的“辛场论”(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而神秘的相似之处。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“辛和接触拓扑学”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，研究接触流形本身的一种方法是考虑以下关系：我们说，如果两个这样的流形形成辛流形的两个独立的边界部分，那么它们是“辛余界”的。两个给定的接触流形是否共界的问题首先帮助我们理解什么样的接触流形可以存在，而辛场理论是我们研究这一问题的最有力的方法之一。因此，本项目的目的是使用该工具和相关工具来尽可能多地了解接触流形上的辛余边关系。由于以前关于这一主题的大多数结果都集中在具有3维边界的4维流形上，所以我们的目标特别是在更高维上获得新的见解。

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.