This conference is planned to take place 23 - 27 April, 2012, at Yale University in New Haven, CT. It will summarize, synthesize, and project into the future, the work of I.I. Piatetski-Shapiro in the area of automorphic forms, especially L-functions. The conference will be of five days duration, with five talks per day. The talks will investigate 5 main themes that reflect Piatetski-Shapiro's main areas of investigation: functoriality and converse theorems, explicit constructions and periods, p-adic L-functions, geometry, and analytic number theory. Each theme will continue throughout the conference. A volume of proceedings, intended to make these results more accessible and applicable, will be produced. Additional information can be found at the conference website: www.math.yale.edu/automorphicforms2012
Automorphic forms are one of the most fascinating and mysterious of the many branches of number theory. Despite their complexity, they have had remarkable and widely varied applications. Results from the theory of automorphic forms were used in the recent proof by A. Wiles and R. Taylor, of "Fermat's Last Theorem" -- the statement that the sum of two perfect nth powers of whole numbers cannot be a perfect n-th power, if the power n is greater than 2 -- a problem that had been unsolved for over 300 years. Automorphic forms and L-functions are deeply related to prime numbers, which are whole numbers larger than one that have no factors except 1 and themselves. They are the building blocks for multiplication of whole numbers. Prime numbers have recently been used decisively in "public key cryptography", which is the basis of secure transactions on the internet. Prime numbers occur very irregularly among all whole numbers, and the subject of their distribution has attracted an immense amount of research. The Riemann zeta function, which offers a path to a refined understanding of the distribution of prime numbers among all positive integers, is the first example of an L-function. L-functions also provide a means of expressing subtle "reciprocity laws" that govern the solutions of polynomial equations. Automorphic forms are also the source of some of the most beautiful formulas in mathematics, for example, Jacobi's formula for the number of ways to express a whole number as a sum of four perfect squares. Finally, there are remarkable connections between the theory of automorphic forms and physics. The same mathematical structure that is foundational to quantum mechanics (the Heisenberg Canonical Commutation Relations), is also the setting for one of the most important methods for constructing automorphic forms. In addition, a variety of formulas from theoretical physics and related mathematics have been discovered in recent years to have strong connections to the theory or automorphic forms. I. I. Piatetski-Shapiro was a world leader in the theory of automorphic forms. The heart of his contributions involved establishing close connections between automorphic forms and L-functions, especially through his "Converse Theorem", which gave detailed conditions for a family of L-functions to be related to an automorphic form. This conference offers an opportunity to synthesize, disseminate, and build on Piatetski-Shapiro's work, to increase our understanding of these fascinating ideas.
", held at Yale University, April 23 - 27, 2012. Automorphic forms are mathematical objects that encode huge amounts of information about phenomena relating to number theory, physics, combinatorics, geometry, and many other seemingly independent fields. They played a critical role in the celebrated proof of Fermat's Last Theorem, and of many other recent striking developments in number theory. They play a key role in the study of lattices in Euclidean spaces, sphere packing, coding theory and structure and classification of finite groups. They are involved in some of the most challenging outstanding conjectures in contemporary mathematics, including Clay Millennium Problems. They have been the subject of intense research in recent decades, substantially (though not entirely) inspired by the Langlands Program. During his long and productive research career, I. I. Piatetski-Shapiro was a leader in producing ideas for studying automorphic forms. The goal of this conference was to review and synthesize the work of Piatetski-Shapiro and collaborators, and to make it available for new generations of researchers. The conference featured high-level talks by 21 leading researchers. It was attended by over 100 mathematicians, including a large contingent of young researchers. The conference proceedings will be published by the American Mathematical Society.