Arithmetic geometry studies some of the most elementary objects in mathematics: Diophantine equations. These are polynomial equations, in finitely many variables, with integer coefficients. Problems involving Diophantine equations are often very simple to state, but fiendishly difficult to solve, with some famous open problems goes back centuries, or even millennia. Part of the difficulty comes from the lack of obvious structure that we can use to try to understand these problems. Many of the great advances in mathematics through the centuries have come from searching for structures behind these problems, and from developing the theoretical tools needed to work with the strucetures that have been found.
My own approach to these problems involves applying tools and techniques from the field of topology. This is a branch of mathematics invented about a century or so ago, and often goes by the name of 'rubber sheet geometry'. The reason for name is that, in topology, we try to understand properties of spaces that are invariant under continuous deformations such as bending, or stretching, or folding. In this world, concrete geometric notions, like distances between points, or angles between lines, no longer have any meaning. More abstract properties of a shape, however, such as having a certain number of holes, or a certain number of ends, or of being finite or infinite in extent, still make sense.
Of course, the Diophantine equations we study in arithmetic geometry are very 'discrete' objects, and cannot be directly thought of as geometric spaces of the kind that topologists study. Instead, the approach taken is indirect, finding different perspectives on these equations, and different ways of understanding them, from which they behave very much like more obviously 'topological' spaces. This way, we can apply the methods of topology to our problem, using geometric intuition and arguing via analogy to discover new structure in the theory of Diophantine equations.
Often, there can be a wide variety of ways of achieving this, even for a single equation or class of equations, and these different approaches will have significant advantages and disadvantages over each other. Some might lend themselves better to concrete calculations, whilst others might be better suited to abstract argumentation. The goal of this project is to try to construct a bridge between some of these different ways of applying topological methods to arithmetic geometry, and therefore combine the relative strengths of each. By fusing concrete computability with abstract machinery, I will be able to provide a more powerful topological toolkit, which we can then use to better understand these oldest of mathematical objects.
