During the Winter and Spring of 2012, the Fields Institute will mount an intensive program on the subject of Galois Representations, Diophantine Equations, and Automorphic Forms, and this will be one of the principal activities of the Institute during that period. The proposed activity will provide support for the participation of early career U.S. mathematical researchers in the program. There has been a very rapid development of ideas and techniques in the area of modularity of Galois representations and its connections with the Langlands program, following the pioneering work of Wiles and Taylor-Wiles on the Shimura-Taniyama conjecture; recent examples include the proof of the Sato--Tate conjecture by Clozel, Harris, Shepherd-Barron, and Taylor, and the proof of Serre's modularity conjecture by Khare, Wintenberger, and Kisin. There is every indication that these results are only the beginning of the potential harvest. The consequences of new progress in the theory of automorphic representations (e.g. the fundamental lemma, proved by Ngo), the emerging theory of a p-adic local Langlands correspondence, the theory of p-adic modular forms, and new ideas in deformation theory and the local structure of Shimura varieties, have only begun to be explored, and significant developments can be expected in the next few years. All of these developments, both existing and potential, will be explored as part of the Fields program, and early career U.S. mathematicians will benefit greatly from the chance to learn about, and indeed to participate in, these developments.
Number theory is the branch of mathematics that studies phenomena related to properties of whole numbers. A typical number theoretic question is to determine the number of whole number solutions of some equation of interest. The answers to such questions can often be encoded in certain mathematical functions known as L-functions. The mathematician Robert Langlands has developed a series of conjectures (or mathematical predictions) regarding L-functions, which predict that any L-function should arise from another kind of mathematical function called an automorphic form. (Number theorists refer to Langlands conjectured relationship between L-functions and automorphic forms as a ``reciprocity law''.) Langlands developed an array of powerful representation theoretic methods to study his conjectures. These are methods that exploit the many symmetries of automorphic forms and L-functions to analyze their mathematical properties; these methods have been incorporated into a body of mathematics known as ``the Langlands program'', which is one of the central areas of modern number theory, and, indeed, of modern pure mathematics. In Winter and Spring of 2012, the Fields Institute will run a thematic program, titled Galois Representations, Diophantine Equations, and Automorphic Forms, which will be dedicated to studying the Langlands program. The present activity will provide support for early career U.S. mathematician to participate in the Fields program, who will thereby be provided with the opportunity both to learn about recent developments in the Langlands program, and to develop the skills necessary to help contribute to the next wave of developments.
" being held at the Fields Institute in Toronto in Winter 2012. The topics of the workshop are central topics in number theory, which itself is one of the core subjects of contemporary pure mathematics, and they formed the key ingredients in the proof of Fermat's Last Theorem by Andrew Wiles roughly twenty years ago. While that proof closed one chapter in the book of mathematics, by proving Fermat's long-outstanding problem, it opened another chapter, since Wiles's proof introduced many new ideas and techniques, which have been taken up by the next generation of number-theorists in order to solve many other outstanding problems. The momentum that has built up in number-theory since Wiles's proof is continuing unabated, and the thematic program highlighted some of the more important recent breakthroughs in the area to a large audience of participating mathematicians. The NSF funding allowed many beginning US-based mathematicians to participate in the program, and helped in transmitting the momentum in the subject from the current generation of leading researchers to these beginning researchers, who will form the next generation of leaders in the field.
