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

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.