Vector bundles and their role in number theory

数论研究整数的算术性质。这是一个深刻而美丽的理论，与数学和物理的许多领域有着惊人的联系。这项建议涉及到研究数字和几何某些方面之间的联系。我们追求的特定连接是通过一种称为模块化形式的功能。模形式在一个多世纪前出现在数学中，在过去的五十年里，它们在数论中的重要性有了很大的增长，这在很大程度上要归功于所谓的朗兰兹计划和安德鲁·怀尔斯的惊人工作，他使用模形式来解决费马最后的定理。模形式是几何空间上的函数(更准确地说，是线丛的部分)，它们本身就是算术上的兴趣。使用传统方法研究模块化形式的部分困难在于导航这些美丽但复杂的空间的几何形状。另一种方法是将几个模形式拼凑成一个向量值函数(更准确地说，是向量丛的一部分)，该函数位于一个看起来像球体的更简单的空间上。也就是说，我们不再考虑复杂空间上的简单函数，而是考虑球面上的复杂函数。这种方法允许我们将新的几何技术(向量丛及其模)引入到模形式的研究中。我们特别感兴趣的是继续我们的探索，即向量丛理论如何影响模形式的结构理论。这项研究对数论的研究人员很重要，他们正开始在朗兰兹计划和现代几何趋势之间建立重要的联系。一个值得注意的例子是Ngo Baao Chau最近对所谓的基本引理的证明，该引理在2010年为Ngo赢得了菲尔兹奖，被《时代》杂志称为2009年第七大最重要的科学发现。证明的关键一步是利用希格斯丛理论，该理论可以追溯到1987年，自那以来一直是几何学的推动力。自这项基础性工作以来，朗兰兹计划中的许多专家已经开始解开希格斯丛如何适用于他们的算术运算。他们正在取得令人振奋的进展，尽管该领域仍然相当年轻。希格斯束在我们的模形式工作中是自然而然地出现的，所以我们对使用这些强大的技术在该领域取得进展的前景感到非常兴奋。我们的研究将扩大前沿几何在数论研究中的应用。我们希望它将有助于普及这一领域，为数论家提供强大的几何工具，并希望它将使更多的几何学家对他们的学科与数论之间的联系感兴趣。

Number theory concerns the study of arithmetic properties of integers. It is a deep and beautiful theory that has surprising connections to many areas of mathematics and physics. This proposal concerns connections between the study of numbers and certain aspects of geometry. The particular connection that we pursue is via a type of function called a modular form. Modular forms appeared in mathematics over a century ago, and their import in number theory has grown considerably in the last fifty years, thanks in large part due to the so-called Langlands program and the stunning work of Andrew Wiles, which used modular forms to solve Fermat's last theorem.Modular forms are functions (more precisely: sections of line bundles) on geometric spaces that are themselves of arithmetic interest. Part of the difficulty in studying modular forms using traditional methods lies in navigating the geometry of these beautiful but complicated spaces. A different approach is to cobble together several modular forms into a single vector valued function (more precisely: section of a vector bundle) that lives on a simpler space that looks like a sphere. That is, instead of thinking about simple functions on a complicated space, we think about complicated functions on a sphere. This approach allows us to introduce new geometric techniques (vector bundles and their moduli) into the study of modular forms. We are particularly interesting in continuing our exploration of how the theory of vector bundles informs the structure theory of modular forms.This research is important to researchers in number theory, who are beginning to make important connections between the Langlands program and modern trends in geometry. A notable example is the recent proof of the so-called Fundamental lemma by Ngo Bao Chau, which earned Ngo a Fields medal in 2010, and which was called the seventh most important scientific discover of 2009 by Time magazine. A key step in the proof was to utilise the theory of Higgs bundles, which dates back to 1987 and which has been a driving force in geometry ever since. Since this fundamental work, a number of experts in the Langlands program have begun to unravel how Higgs bundles fit into their arithmetic work. They are making exciting progress, even though the field is still rather young. Higgs bundles are arising naturally in our work on modular forms, and so we are very excited by the prospect of using these powerful techniques to make progress in the field. Our research will expand this use of cutting edge geometry in the study of number theory. We hope that it will help popularise this area, that it will give number theorists powerful geometric tools, and that it will interest more geometers in the connections between their subject and number theory.