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，与数学和物理学的许多领域有着惊人的联系。这个建议涉及数字研究和几何某些方面之间的联系。我们追求的特殊联系是通过一种叫做模形式的函数。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显著的增长，这在很大程度上要归功于所谓的朗兰兹纲领和安德鲁·怀尔斯的惊人工作，他用模形式解出了费马大定理。模形式是几何空间上的函数（更准确地说：线束的部分），它们本身具有算术意义。使用传统方法研究模形式的部分困难在于如何驾驭这些美丽而复杂的空间的几何形状。另一种不同的方法是将几个模块形式拼凑成一个单一的矢量值函数（更准确地说：矢量束的一部分），它存在于一个看起来像球体的简单空间中。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（向量束及其模）引入到模形式的研究中。我们特别感兴趣的是继续探索向量束的理论如何影响模形式的结构理论。这项研究对数论研究人员很重要，他们开始在朗兰兹纲领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吴宝洲最近对所谓的基本引理的证明，他在2010年获得了菲尔兹奖，并被《时代》杂志评为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束理论，该理论可以追溯到1987年，自那以后一直是几何学的推动力。自从这项基础性工作以来，朗兰兹计划的许多专家已经开始揭示希格斯束是如何与他们的算术工作相适应的。尽管这个领域还很年轻，但他们正在取得令人兴奋的进展。希格斯束是在我们对模形式的研究中自然产生的，因此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。我们的研究将在数论研究中扩展这种前沿几何的应用。我们希望它能帮助推广这一领域，给数论学家提供强大的几何工具，并使更多的几何学家对他们的学科和数论之间的联系产生兴趣。

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.