Vector bundles and their role in number theory

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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.