Arithmetic and topology of moduli spaces

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The mathematical landscape is divided into three big areas: algebra, analysis and geometry. Algebraic geometry is situated between algebra and geometry, and is concerned with the study of solutions of systems of polynomial equations in several variables. As the number of solutions is usually infinite, one can rarely compute all of them, and instead contents oneself with an understanding of the geometric shape formed by the ensemble of solutions. This is what we call an algebraic variety. It is the principal object of interest in algebraic geometry. Applications of this theory are wide-ranging: a special case, known as elliptic curves, plays a crucial role in the implementation of modern cryptography systems. Algebraic geometry also prominently appears in physics: string theory predicts that the four dimensions of space and time are supplemented by 6 extra dimensions which are curled up in the shape of a tiny algebraic variety (known as Calabi-Yau varieties). This project is devoted to the study of algebraic varieties known as moduli spaces. Rather than arising as the set of solutions to an explicit system of equations, each point of a moduli space represents a fixed type of mathematical objects, and describes its variations and deformations. Due to this geometric interpretation, the theory of moduli spaces is extremely rich and leads to beautiful applications in other areas of mathematics. We will study moduli spaces arising in algebraic geometry through tools provided by number theory. This allows us to confirm predictions originating in mathematical physics, in particular string theory. One of the main protagonists in my research is the moduli space of Higgs bundles. Together with my collaborators Dimitri Wyss and Paul Ziegler we proved a conjecture by Hausel and Thaddeus which relates the geometry of two such moduli spaces, related by Langlands duality. Their prediction was heavily influenced by mirror symmetry (a phenomenon observed in string theory), but our proof provides an entirely arithmetic approach. In a sequel to our proof of the Hausel-Thaddeus conjecture, we reverse the flow of ideas: our methods are used to give a new and elementary proof of the fundamental lemma (proven by Ngô in 2008). The latter is central to the Langlands programme, and hence to modern day understanding of number theory. In future work I will continue to explore this exciting connection between number theory, algebraic geometry and physics.