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

几何中的基本实体是欧几里得n-空间。这是一个用n个坐标来描述你的位置的空间，其中n是一个正整数。我们熟悉的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。并非所有几何体都是平坦的：以球体或甜甜圈的表面为例。然而，如果我们站在一个球体上，它像地球一样大，那么它对我们来说就像平坦的欧几里得2空间，至少在附近是这样。因此，球面是一个流形：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，地球内部是一个三维流形。我的课程是微分几何，这是一门研究流形的学科。想象一下，你有一个网球，你在上面画了一个赤道。赤道是一个位于球上的圆。由于圆是一维流形，因此赤道是球表面的一个子流形，也就是说，它是一个位于更大流形中的流形。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，我们可以认为流体流动和重力是关于流形几何的额外信息。有一项数据被称为特殊完整群，它只能发生在七维和八维中；这使得这些维特别令人着迷。具有特殊完整群的流形称为七维G_2流形和八维自旋(7)流形。我的工作是关于特殊的四维子流形，称为G_2流形中的共结合4-折叠和自旋(7)流形中的Cayley 4-折叠。CoAssociative和Cayley 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，它们收缩是为了在受到约束的情况下使其表面积最小，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，但假设我们看的是圆锥体。圆锥体的尖端并不光滑：这是奇点的一个例子，奇点是流形上的一个“坏”点。锥体的另一个性质是它是由它的横截面定义的。如果我们把圆锥体的尖端和球体的中心放在同一位置，那么圆锥体与球体表面相交的点集称为圆锥体的连环。连杆是圆锥体的横截面和球面的子流形。一般而言，我们首先将n-球面定义为欧几里得(n+1)-空间中距原点都有单位距离的点集。然后，如果我们在欧几里得(n+1)空间中有一个四维锥体，它的链接是n球体的三维子流形。我研究的一个令人兴奋的方面是它与一个被称为弦理论的物理领域的联系。这一理论试图通过将粒子视为“线”而不是点来描述宇宙的运作方式。这个想法的一个奇怪的副产品是，宇宙必须有很多维度。具体地说，我们必须将宇宙想象成有10维、11维或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维碎片组成；这就是为什么它与我的工作有关。我想要解决的第一个问题是找到用共结合或Cayley 4折叠覆盖G_2或Spin(7)流形的方法，这些流形可能有奇点，使得流形上的每个点只覆盖一次。这些解将有助于回答弦理论中的难题。理解奇点是几何的重要组成部分。我的项目的另一部分是发现哪些锥形奇点可能发生。要做到这一点，我想找出一个三维流形什么时候可以被推到6球或7球中，使它成为一个子流形，它是余结合锥或Cayley锥的连接。

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.