The main focus of the research project concerns the moduli space of representations of the fundamental group of a smooth projective algebraic curve to a reductive algebraic group G. This moduli space is the Betti version of a cohomology group in non-abelian Hodge theory. Its other versions are, Dolbeault: the moduli space of semistable Higgs G-bundles on the surface and, de Rham: the moduli space of flat G-connections on it. In the Dolbeault and de Rham versions this space has been central to recent important work in the Langlands program, both the arithmetic by work of Ngo and Laumon and the geometric by work of Kapustin and Witten. The basic idea of the project is to use tools of Number Theory, Combinatorics and the Representation Theory of finite groups of Lie type to count points on the Betti space over finite fields. The Weil conjectures then yield cohomological and geometrical information about it (and hence also about its other two flavors).
In the long run, Mathematics always has a way of making itself useful outside its own discipline. Who would have thought 20 years ago, say, that something as apparently removed from everyday life as the concept of an "elliptic curve" could end up as a crucial tool for the security of online shopping? An elliptic curve is a geometric construct with a very rich structure. Two points on the curve can be "added" by geometric means to produce a third point. This fact goes back more than 300 years. In more modern times we have learned how do to "geometry" (work with points, lines, curves, etc.) in a purely finite context, where the real or complex numbers are replaced by their finite counterparts, finite fields. This provides one of the most useful ways to apply abstract mathematical concepts from Geometry and Number Theory to real life situations. The research project uses finite fields in a different way: to probe the geometry, in the usual sense of the word, of certain spaces of great interest to both Mathematics and Physics. The PI finds the interplay between the discrete (counting points over finite fields) and the continuous (geometry over the complex numbers) in this project, as well as the fact that involves in a substantial way fairly distant areas of Mathematics (Number Theory, Combinatorics, Group Theory and Differential Geometry), fascinating.
At its core this project is about the interaction between Geometry (the continuous) and Number Theory (the discrete). Traditionally Geometry deals with forms and shapes that come from the world as we experience it. We use it to solve practical problems involving angles, distances, curves, surfaces, etc. Within Mathematics, however, Geometry took a life of its own, independent of its everyday-world underpinnings. In Algebraic Geometry we are interested in solutions to algebraic equations. For example, the set of complex numbers x,y satisfying the equation y^2+x^5+1=0. We can visualize these solutions as two donuts glued together. We say that this forms an algebraic curve of genus two. In Number Theory we may replace equality by the congruence modulo a prime p. I.e., we now look for x,y from among the integers 0,1,...,p-1 such that y^2+x^5+1 is divisible by p. In this context 'equal to zero' is replaced by 'is divisible by p'. In technical terms we have replaced the field of complex numbers by a finite field. From the point of view of algebra, numbers in a finite field behave very much like the real or complex numbers. What kind of geometry can this lead to? Though not so easy to visualize, it leads to a geometry that still retains a strong connection to the more familiar one. And it comes with a new tool: we can count. The fact that our curve has genus 2, for example, manifests itself in how the number of solutions to grows as we vary the finite field. This connection between the finite field and complex geometries gives rise to the possibility of using one to infer facts about the other. This has proved to be extremely useful and is, in a nutshell, the underlying principle behind the present research project. We use the counting of solutions over finite fields to obtain results about the complex geometry of the moduli space of Hitchin pairs. This space is of central significance in Differentiable Geometry with deep rooted connections to modern Physics. We should add that in a amazing turnaround finite field geometry now plays a fundamental role in our everyday life. Indeed, the current standard algorithm used for cryptography (recommended by the National Security Agency) involves elliptic curves, genus one algebraic curves, over finite fields. In other words, finite field geometry is likely to be in the background every time we shop online or chek our e-mail from home. By using the connection with finite field geometry my collaborators, Tamas Hausel and Emmanuel Letellier, and I managed to establish a number of non-trivial results on the geometry of the Hitchin spaces. For example, we gave explicit formulas for their E-polynomial (a geometric invariant of the space). As an unexpected bonus, our methods allowed us to complete the proof of a 30 year old conjecture in the field of quiver representations due to V. Kac. Moreover, our research led to several new conjectures in the field and the inroads made are still fertile ground for further work. The rough timeline of this award was as follows. - We spent, with my collaborators Hausel and Letellier, two weeks in April 2008 as part of the Research in Pairs program at the Mathematisches Forschungsinstitut Oberwolfach in Germany. This intense research activity was crucial to the development of the project. - I was a Visiting Full Professor for the month of June 2009 at the Universite de Caen, France (where Letellier is in the Faculty). - I co-organized with Letellier and O. Schiffmann (from the Ecole Normal Superieure, Paris) a workshop at the Clay Mathematical Institute in Cambridge, MA in May 2009 "Macdonald Polynomials and Geometry". This 4-day workshop fully funded the participation of about 15 researchers. Macdonald polynomials are central to modern combinatorics in particular to its relation to Algebraic Geometry. They also play a fundamental role in our project and hence the reason for this workshop. - I spent a sabbatical year in 2009-2010 at Oxford University visting Tamas Hausel. - Letellier organized a mini-workshop in Caen in May 2012 centered around our joint work. In 2010 four of my students completed their Ph.D. under my supervision: Silvia Aducci, Kim Hopkins, Miguel Rodriguez and Martin Mereb. During the time of this award Martin Mereb went from being my Ph.D. student to a postdoctoral position at Oxford University and now at EPFL in Lausanne. His thesis was very close to the subject of this award and together with Hausel we are close to finishing an important project. During the period of this award I gave two talks in the Mathemtics Teacher Circle of Austin. This activity is geared to local middle-school teachers and is quite successful. I was a judge for the regional Siemens high-school competition in two different opportunities. I also co-organized other academic activities such as the Arizona Winter School on quadratic forms in 2009 (a large activity for graduate students in Mathematics held annually in Tucson, Arizona funded by the NSF) and a workshop on L-functions in Benasque, Spain. In addition, I gave several academic presentations of the work around the world; e.g., Colloquium in Madrid, Spain in 2010; Seminario de Geometria Algebraica, Universidad de Buenos Aires, Argentina, 2011; Colloquium, University of Edinburgh, Scotland, 2010; Oberwolfach Mathematics Institute, Germany 2010 and 2011.
