A big theme in number theory in the last 50 years has been the relationship between automorphic forms, Galois representations and objects from algebraic geometry. There is an extensive web of extraordinary conjectures (for instance the Artin conjecture, the Shimura-Taniyama conjecture, Langlands' conjectures, Serre's conjecture and the Fontaine-Mazur conjecture) linking these three seemingly very different subjects (which relate to analysis, algebra and geometry respectively). Progress on these conjectures is currently very exciting (particularly recent work of Kisin and Khare-Wintenberger). The PI proposes (with K. Buzzard and T. Gee) to prove the Artin conjecture for odd, two dimensional representations of the absolute Galois group of the rational numbers. He also proposes to investigate generalisations of Ihara's lemma to higher rank unitary groups. As the PI (with L. Clozel, M. Harris and N. Shepherd-Barron) has recently shown, this would imply the Sato-Tate conjecture for rational elliptic curves with multiplicative reduction somewhere.
This circle of ideas is the one that led to Andrew Wiles' celebrated proof of Fermat's last theorem after over 300 years. They fall into the general area of arithmetic geometry - a subject that blends two of the oldest areas of mathematics: number theory and geometry. This combination has proved extraordinarily fruitful. Among its many consequences are new error correcting codes. Such codes are essential for both modern computers (hard disks) and compact disks.
The research carried out under this grant concerns the deep relationships between very different mathematical objects: automorphic forms, Galois representations and motives. Automorphic forms are objects connected with the symmetries of various non-standard geometries, for instance the strangely curved hyperbolic plane which is illustrated in some of Escher's woodcuts. Galois representations are, on the other hand, connected with the (algebraic) symmetries enjoyed by the solutions to polynomial equations. Motives are also algebraic objects. The sort of mathematics that has gone into the study of geometric symmetries has been very different from the sort of mathematics that has gone into the study of symmetries of algebraic objects. Nevertheless in the last 60 years it has been discovered that there seemed to be an extraordinary relationship, a sort of duality, between these two branches of mathematics. A big theme in mathematics over these 60 years, the Langlands program, has been to prove that these relationshps really exist. When one can establish a highly nontrivial dictionary of this sort between two very different fields it can have a big impact, as problems that are hard in one field can often be translated to problems that are easy in the other. For instance progress on these problems led to Andrew Wiles' celebrated proof of Fermat's last theorem over 300 years after the problem was first posed. Wiles turned a very hard algebraic problem (Fermat's Last Theorem) into an easy problem about the symmetries of the hyperbolic plane (the non-existence of modular forms of weight 2 and level 2). This field is also related to the development of new error correcting codes which are essential to the accurate transmaission of large amounts of data for instance via CD's, DVD's or satellite links. Beginning with Wiles' work in the early 1990's a great deal of progress was made on these problems in the simplest, two dimensional, case. The biggest achievement of the research conducted under this grant was to extend much of this work to the general case (any dimension). With Barnet-Lamb, Gee and Geraghty we proved that any regular, self-dual motive is potentially automorphic (i.e. occurs in the dictionary I mentioned above). We made several other related innovations, including the proof of the Sato-Tate conjecture for all non-CM elliptic curves over totally real fields. As part of the grant I helped to train 12 graduate students and 6 post-docs in these areas of mathematics. 8 of the graduate students obtained PhD's, 2 obtained masters degrees and 2 are still working towards their PhD. One of these recent PhDs now has a tenure track job at MIT, one at the University of Minnesota, one at Boston College, one has an extremely prestigous Clay five year fellowship, two have NSF post-docs at Princeton University, one has a tenured job at KAIST in Korea and one left academia to work for a Hedge fund.
