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.******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.