An overconvergent Riemann-Hilbert correspondence

算术几何研究数学中一些最基本的对象：丢番图方程。这些是多项式方程，有许多变量，系数为整数。涉及丢番图方程的问题通常很容易陈述，但很难解决，一些著名的公开问题可以追溯到几个世纪，甚至几千年前。部分困难来自于缺乏我们可以用来理解这些问题的明显结构。几个世纪以来，数学的许多重大进步都来自于对这些问题背后的结构的探索，以及对已经发现的结构所需的理论工具的开发。我自己解决这些问题的方法涉及到应用拓扑学领域的工具和技术。这是数学的一个分支，发明于大约世纪以前，通常被称为“橡胶片几何”。命名的原因是，在拓扑学中，我们试图理解空间在连续变形（如弯曲、拉伸或折叠）下不变的性质。在这个世界上，具体的几何概念，如点之间的距离，或线之间的角度，不再有任何意义。然而，形状的更抽象的性质，例如具有一定数量的孔，或一定数量的端部，或在范围上是有限的或无限的，仍然是有意义的。当然，我们在算术几何中研究的丢番图方程是非常“离散”的对象，不能直接认为是拓扑学家研究的那种几何空间。相反，所采取的方法是间接的，找到不同的角度对这些方程，和不同的方式来理解他们，从他们的行为非常像更明显的“拓扑”空间。这样，我们就可以将拓扑学的方法应用到我们的问题中，使用几何直观和通过类比来论证，以发现丢番图方程理论中的新结构。通常，即使对于单个方程或一类方程，也可以有各种各样的方法来实现这一点，这些不同的方法彼此之间会有显着的优点和缺点。有些可能更适合具体的计算，而另一些可能更适合抽象的论证。这个项目的目标是试图在这些不同的将拓扑方法应用于算术几何的方法之间建立一座桥梁，从而联合收割机结合各自的相对优势。通过将具体的可计算性与抽象的机器融合在一起，我将能够提供一个更强大的拓扑工具包，然后我们可以用它来更好地理解这些最古老的数学对象。

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.