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。

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.