Applications of Higher Dimensional Algebra to Stable Homotopy Theory

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Category theory is the abstract study of relationships between mathematical objects. These relationships can be simple, such as one number being larger or smaller than another, or they can be more complicated and many-layered, such as spatial relationships between two points on a surface. Higher-dimensional category theory is then the study of similar kinds of relationships, but we now allow the existence of relationships between relationships, and so on. One example might describe all the ways to walk from one point to another. If the space between these two points is an empty field, say, then any two different paths are essentially the same, as we could devise a series of paths that gradually changed from the first given path to the second. On the other hand, if there is a pond between these two points, then there are at least two essentially different paths, one for each way around the pond. The higher dimensional category theory in this proposal is in the subfield of higher dimensional algebra which is the study of objects with many different kinds of algebraic operations on them, and the interactions between those; a simple example would just be the real numbers with the operations plus and times, and one example interaction between these is a(b+c)=ab+ac.The other field of mathematics in this proposal is topology, which is the study of shapes. In topology, aspects like distance or smoothness do not matter, so a circle is the same as a square and a doughnut is the same as a coffee cup. One of the most profitable ways of understanding shapes is by assigning them algebraic invariants. For instance, there is an invariant called the fundamental group which counts certain kinds of holes in a shape; computing the fundamental group of both a circle and a square will give the same answer, as they both have a single hole in the middle. On the other hand, computing the fundamental group of a filled-in square will give a different answer, precisely because the hole has now been filled.The research in this proposal is about using higher dimensional algebra to describe shapes. As an example, imagine two flexible tubes. We can combine these in a variety of ways: we can just set them next to each to get a pair of tubes, we could glue the end of one tube to the other and get a single very long tube, or we could even glue both ends of the first tube to the corresponding ends of the second tube to get a circular tube (a shape called a torus). Gluing surfaces together is one kind of algebraic operation, and setting surfaces next to each other is another, and these two operations behave in ways that are governed by laws appearing in higher dimesional algebra. If we additionally take into account symmetries of the shapes involved (for our example, tubes can be rotated or the ends can be swapped), then both the topological and algebraic descriptions become more complicated. These constructions have been studied using topology, but this research is aimed at utilising the tools of higher dimensional algebra in order to shed new light on old problems.