The month-long conference, "Rational Points, Rational Curves, and Entire Holomorphic Curves on Varieties", June 3-28, 2013, will be held at the Centre de Recherches Mathematiques in Montreal, Quebec, Canada. The URL for the conference website is as follows. www.crm.umontreal.ca/2013/Integral13/index_e.php
The proposed activity is a summer school of 2-3 weeks comprised of a dozen mini-courses, followed by a week-long international workshop. There have been recent advances in seemingly separate fields: classification of complex projective varieties, existence and properties of solutions of Diophantine equations, and analytic behavior of entire curves in projective varieties. The moment is ripe for a summer school and workshop bringing together experts to synthesize these results, and train junior mathematicians in the new techniques. With the groundwork laid, we will emphasize open problems that seem amenable to solution: the holomorphic Lang conjecture about entire curves in general type varieties, potential density of rational points and rational curves on Calabi-Yau varieties, and the Grothendieck-Katz conjecture on algebraic solutions of differential equations.
Systems of polynomial equations arise in all areas of mathematics, as well as areas of science and engineering as disparate as genomics and robot design. The oldest, and still most important, problem in this area is that of finding solutions to polynomial systems, particularly solutions that are fractions of whole numbers. Although surprising, this problem is closely connected to the related problem of interpolating any two given solutions by a system of solutions that are the outputs of a fraction of polynomials, or, more generally, an entire analytic function (the functions most amenable to study via "power series expansion"). There have been recent breakthroughs on both of these problems separately, as well as the interaction between them. This conference will feature training courses for young mathematicians, discussions with experts, and a public outreach lecture, as well as a planned volume to disseminate this work to an even broader audience.
Historically, two of the most influential problems in the mathematics of systems of polynomial equations are the "Diophantine problem" and the "parameterization problem". The Diophantine problem asks, under the assumption that the polynomial equations have whole number coefficients (or, more generally, the coefficients lie in a "global field"), whether or not there exists a (simultaneous) solution of the polynomial equation in whole numbers. This problem is vital in number theory, particularly in cryptographic applications. The parameterization problem asks, is there a way to parameterize all solutions (real number solutions or complex number solutions) as the outputs of a polynomial function, i.e., a simple machine that produces all solutions as outputs by varying the inputs arbitrarily. This problem is vital in applied mathematics and engineering applications of polynomial equations, e.g., design of robots to perform a sequence of specified tasks. Although the problems appear completely different, they turn out to be intimately linked via the geometry of the set of all (complex number) solutions, and this connects the number-theoretic and polynomial problems to complex analysis. This project was a one-month training event and research conference bringing together both experts and junior participants from these three fields -- number theory, algebraic geometry and complex analysis -- to report on the many recent advances between these disciplines and to learn the key ideas that might be used to make further progress. There were expository lectures for students, problem sessions, research reports by experts, and a special lecture for the general public. There is a conference volume underway, collecting lecture notes, general surveys and research reports, to be published in the near future.
