CR Geometry is a developing branch of mathematics which arose from the theory of functions of several complex variables and which touches nearly all fields of mathematics. The name itself has two etymologies: CR stands for Cauchy-Riemann and suggests the Cauchy-Riemann equations; CR also stands for complex-real and suggests real submanifolds of complex spaces. The ideas developed together since the 1960's when mathematicians used the methods of partial differential equations (especially the Cauchy-Riemann equations in several variables) to relate the geometry of the boundary of a domain in complex Euclidean space to the function theory on the domain. These considerations led to a deep understanding of pseudoconvexity, a complex variables analogue of convexity, and to the study of the tangential Cauchy-Riemann equations. This summer's workshop will have three parts. We will offer a short course on CR geometry that should serve as useful background for research in many areas. We will present some accessible research problems based on CR mappings and we will work together on them. Some of these problems involve elementary number theory. For example, CR geometry leads naturally to a family of polynomials with integer coefficients in two variables with the remarkable property that f(x,y) is congruent to x^p + y^p modulo (p) if and only if p is prime. The simplest example is the familiar polynomial(x+y)^p. We will work on open questions concerning them. Another open problem we will consider has a completely different flavor; it involves iterated Lie brackets and could lead to a new characterization of pseudoconvexity. We will conclude with a small conference where research mathematicians give lectures and meet the graduate students. The main lectures will be given by John Dâ€™Angelo; the following have agreed to participate in the program: Peter Ebenfelt (UCSD), Dror Varolin (UIUC), Jeff McNeal (Ohio State), Bernhard Lamel (Univ. of Vienna), Xiaojun Huang (Rutgers). Students interested in this program might consult the references below to get a feeling for CR geometry. On the other hand, what happens in the workshop depends mostly on the imagination of the students. This is an MSRI Summer Graduate Program; priority is given to students nominated by MSRI's Academic Sponsor universities. Funding is available only for nominee students. References M. Salah Baouendi, Peter Ebenfelt, and Linda Preiss Rothschild, Real submanifolds in complex space and their mappings, Princeton Mathematical Series 47, Princeton University Press, Princeton, NJ, 1999. Tom Bloom, David Catlin, John P. D'Angelo and Yum-Tong Siu, editors, Modern Methods in Complex Analysis, Annals of Math Studies 137, Princeton Univ. Press, Princeton, 1995. John P. D'Angelo, Several Complex Variables and the Geometry of Real Hypersurfaces, CRC Press, Boca Raton, Fla., 1992. John P. D'Angelo, Inequalities from Complex Analysis, Carus Mathematical Monograph No. 28, Mathematics Association of America, 2002. John P. D'Angelo, Number-theoretic Properties of Certain CR Mappings, Journal of Geometric Analysis, Volume 14, Number 2 (2004), 215-229.
Summer Graduate School