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The Exceptional Geometry of Seven and Eight Dimensions: Coverings and Four-Dimensional Cones

七维和八维的特殊几何：覆盖物和四维锥体

基本信息

批准号：
EP/H003584/1
负责人：
Jason Lotay
金额：
$ 56.65万
依托单位：
Imperial College London
依托单位国家：
英国
项目类别：
Fellowship
财政年份：
2009
资助国家：
英国
起止时间：
2009 至无数据
项目状态：
已结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FH003584%2F1
关键词：
Exceptional Geometry Seven Eight Dimensions

项目摘要

The basic entity in geometry is Euclidean n-space. It is a space where you describe your position using n coordinates, where n is a positive whole number. We are familiar with n is 1, 2 and 3: these are the straight line, the flat plane, and the usual 3-dimensional space with x, y and z-axes respectively. Not all geometry is flat: take the surface of a sphere or a doughnut, for example. However, if we stand on a sphere, and it is large like the Earth, then it looks like flat Euclidean 2-space to us, at least close by. Thus, the surface of a sphere is a manifold: a shape which looks like Euclidean n-space near each point, but is not necessarily flat. The surface of a doughnut is also a 2-dimensional manifold and the interior of the Earth is a 3-dimensional manifold. My subject is Differential Geometry, which is the study of manifolds.Imagine you have a tennis ball and you draw an equator on it. The equator is a circle which lies on the ball. Since a circle is a 1-dimensional manifold, the equator is a submanifold of the surface of the ball; that is, it is a manifold sitting inside a bigger manifold. My research is all about submanifolds.You can do a lot with manifolds by putting more geometric structure on them. For example, we can think of fluid flow and gravity as extra information about the geometry of a manifold. One piece of data is called an exceptional holonomy group which can only happen in dimensions seven and eight; this makes these dimensions particularly fascinating. Manifolds with an exceptional holonomy group are called G_2 manifolds in seven dimensions and Spin(7) manifolds in eight. My proposed work is on special 4-dimensional submanifolds called coassociative 4-folds in G_2 manifolds and Cayley 4-folds in Spin(7) manifolds. Coassociative and Cayley 4-folds satisfy equations which mean their area is as small as possible. Therefore, they are like bubbles, which shrink in order to minimize their surface area subject to constraints, such as containing a fixed volume of air.So far we have thought about smooth objects, but suppose we look at a cone. A cone is not smooth at its tip: this is an example of a singularity, which is a 'bad' point on a manifold. Another property of a cone is that it is defined by its cross-section. If we put the tip of a cone and the centre of a sphere at the same place, then the set of points where the cone meets the surface of the sphere is called the link of the cone. The link is a cross-section of the cone and a submanifold of the surface of the sphere. To generalise, we first define the n-sphere as the set of points in Euclidean (n+1)-space which are all unit distance from the origin. Then, if we have a 4-dimensional cone in Euclidean (n+1)-space, its link is a 3-dimensional submanifold of the n-sphere.An exciting aspect of my research is its connection with an area of physics called String Theory. This theory tries to describe how the universe works by thinking of particles not as points, but loops of 'string' instead. A strange by-product of this idea is that the universe has to have many dimensions. Specifically, we have to visualise the universe as having 10, 11 or 12 dimensions, consisting of a large 4-dimensional manifold and a very small extra 6, 7 or 8-dimensional piece; this is why it relates to my work. The first problems that I want to solve are to find ways of covering G_2 or Spin(7) manifolds using coassociative or Cayley 4-folds, which may have singularities, such that every point of the manifold is covered only once. The solutions would help answer difficult questions in String Theory.Understanding singularities is an important part of geometry. The other part of my project is to discover which cone-like singularities can occur. To do this, I want to find out when a 3-dimensional manifold can be pushed into the 6-sphere or the 7-sphere so that it becomes a submanifold which is the link of a coassociative or Cayley cone.

几何中的基本实体是欧几里德n空间。在这个空间中，你用n个坐标来描述你的位置，其中n是一个正整数。我们熟悉n = 1、2和3：它们分别是直线、平面和通常的三维空间，分别是x、y和z轴。并非所有的几何形状都是平坦的：以球体或甜甜圈的表面为例。然而，如果我们站在一个像地球一样大的球体上，那么它对我们来说就像平坦的欧几里得2空间，至少在近处是这样。因此，球体的表面是一个流形：在每个点附近看起来像欧几里得n空间的形状，但不一定是平坦的。甜甜圈的表面也是一个二维流形，而地球的内部是一个三维流形。我的专业是微分几何，也就是流形的研究。想象你有一个网球，你在上面画一个赤道。赤道是在球上的一个圆。由于圆是一维流形，赤道是球表面的子流形；也就是说，它是一个流形在一个更大的流形里。我的研究都是关于子流形的。你可以通过在流形上加入更多的几何结构来做很多事情。例如，我们可以把流体流动和重力看作是流形几何的额外信息。一个数据被称为例外完整群，它只可能发生在第7维和第8维；这使得这些维度特别迷人。具有特殊完整群的流形在7维称为G_2流形，在8维称为Spin(7)流形。我提出的工作是关于特殊的四维子流形，称为G_2流形中的协关联4-折叠和Spin(7)流形中的Cayley 4-折叠。协联折叠和凯莱折叠满足的方程意味着它们的面积尽可能小。因此，它们就像气泡一样，在约束条件下，比如含有固定体积的空气，它们会收缩，以使表面积最小化。到目前为止，我们已经考虑了光滑的物体，但假设我们看一个锥体。圆锥的尖端是不光滑的：这是奇点的一个例子，奇点是流形上的一个“坏”点。圆锥的另一个性质是它是由它的横截面来定义的。如果我们把圆锥体的尖端和球体的中心放在同一个地方，那么圆锥体与球体表面相遇的一组点就叫做圆锥体的连杆。连杆是圆锥的横截面和球体表面的子流形。为了推广，我们首先将n球定义为欧几里德（n+1）空间中距离原点单位距离的点的集合。然后，如果我们在欧氏（n+1）空间中有一个四维锥，它的连杆是n球的一个三维子流形。我的研究中一个令人兴奋的方面是它与一个叫做弦理论的物理学领域的联系。这个理论试图通过把粒子想象成“弦”的环而不是点来描述宇宙是如何运作的。这个想法的一个奇怪的副产品是宇宙必须有多个维度。具体来说，我们必须把宇宙想象成有10、11或12个维度，由一个大的4维流形和一个非常小的额外的6、7或8维部分组成；这就是为什么它与我的工作有关。我要解决的第一个问题是找到覆盖G_2或自旋(7)流形的方法使用协共轭或凯利4折叠，它们可能有奇点，使得流形的每个点只覆盖一次。这些解决方案将有助于回答弦理论中的难题。理解奇点是几何学的重要组成部分。我项目的另一部分是发现哪些锥状奇点可以出现。为了做到这一点，我想找出一个三维流形什么时候可以被推入六球或七球，这样它就变成了子流形，也就是协共轭或凯莱锥的连杆。