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

几何学中的基本实体是欧几里得n-空间。它是一个用n个坐标来描述你的位置的空间，其中n是一个正整数。我们熟悉n是1、2和3：它们分别是直线、平面和通常的具有x、y和z轴的三维空间。并不是所有的几何体都是平的：例如，球体或甜甜圈的表面。然而，如果我们站在一个球体上，并且它像地球一样大，那么它看起来就像平坦的欧几里得2-空间，至少在附近。因此，球面是一个流形：一个在每个点附近看起来像欧几里德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。我的学科是微分几何，这是流形的研究。想象一下，你有一个网球，你在上面画一个赤道。赤道是一个位于球上的圆。由于圆是一维流形，赤道是球表面的子流形;也就是说，它是一个位于更大流形内部的流形。我的研究都是关于子流形的。你可以通过在流形上添加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关于流形几何的额外信息。其中一个数据被称为特殊的完整组，它只能发生在第七维和第八维中;这使得这些维度特别迷人。具有特殊完整群的流形在七维中称为G_2流形，在八维中称为Spin（7）流形。本文主要研究G_2流形中的余结合4-折叠子流形和Spin（7）流形中的Cayley 4-折叠子流形。共缔合和凯莱4-折叠满足方程，这意味着它们的面积尽可能小。因此，它们就像气泡一样，会收缩，以使其表面积最小化，这是受到约束的，例如包含固定体积的空气。到目前为止，我们已经考虑了光滑的物体，但假设我们看一个圆锥体。一个圆锥在它的顶端是不光滑的：这是一个奇点的例子，它是流形上的一个“坏”点。圆锥体的另一个性质是它由它的横截面定义。如果我们把一个圆锥的顶点和一个球面的中心放在同一个地方，那么圆锥与球面相交的点的集合就叫做圆锥的链环。链接是圆锥的横截面和球面的子流形。为了推广，我们首先将n-球面定义为欧几里得（n+1）-空间中所有离原点单位距离的点的集合。那么，如果我们在欧几里得（n+1）空间中有一个四维锥，它的连接是n球面的一个三维子流形。我的研究的一个令人兴奋的方面是它与一个叫做弦论的物理学领域的联系。这个理论试图描述宇宙是如何工作的，它不是把粒子看作点，而是把粒子看作一圈一圈的“弦”。这个想法的一个奇怪的副产品是宇宙必须有许多维度。具体地说，我们必须把宇宙想象成有10、11或12维，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。我想解决的第一个问题是找到用余结合或Cayley 4-折叠覆盖G_2或Spin（7）流形的方法，它们可能有奇点，使得流形的每个点只被覆盖一次。这些解将有助于解答弦论中的难题。理解奇点是几何学的重要组成部分。我的项目的另一部分是发现哪些锥状奇点可能发生。为了做到这一点，我想知道什么时候一个三维流形可以被推进到6-球面或7-球面，使得它成为一个子流形，它是一个余结合锥或凯莱锥的链接。

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.