This grant will support a program in the Fall of 2012 at the Institut Henri Poincare in Paris, France, on the theme of Conformal and Kahler geometry. The trimester will begin with two introductory mini-courses followed by the week-long conference Conformal and Kahler Geometry (September 17-21), which will cover the broad mathematical themes of the program: conformal invariants, Poincaré-Einstein metrics, self-duality, nonlinear PDEs in conformal geometry, critical Kahler metrics, and complex Monge-Ampere equations. The other activities throughout the trimester will attempt to elaborate and bring into sharper focus the developments related in the introductory conference. There will be three smaller and more focused workshops later in the term: Geometry and Physics (October 8-12), Geometric PDE (November 5-9), and Recent Developments in Kahler Geometry (December 10-14). These will be accompanied by a series of topical mini-courses.
Conformal geometry has its origins in the study of functions of a complex variable in the nineteenth century. Roughly speaking, it is the study of transformations of a space which may distort the measurement of distances, but not how one measures angles. Conformal mappings play a basic role in many applied sciences, and modern conformal geometry has connections to theoretical physics and materials science. Kahler geometry is a branch of complex geometry which also has important ties to physics and many areas of modern geometry. The goals of this program are to provide an overview of the current state of the field in these areas, foster interactions between researchers and educate graduate students and postdoctoral scholars on open problems and recent techniques.
" provided support for some US-based participants to attend a three-month long scientific meeting at the Institut Henri Poincare in Paris, France. The mathematical theme of the meeting was recent developments in two areas of modern geometry: conformal and Kahler. Kahler geometry in particular is an area with connections to many other areas of mathematics, including algebraic geometry and mathematical physics (indeed many fundamental ideas and objects in string theory are derived from Kahler geometry). Conformal geometry is a more classical subject, but has deep connections to the theory of differential equations and also the mathematical theory of general relativity. The program took place from 10 September to 14 December, 2012. The Organizing Committee consisted of Matthew Gursky (Notre Dame, USA), Emmanuel Hebey (Cergy-Pontoise), Frank Pacard (Ecole Polytechnique), and Jeff Viaclovsky (Wisconsin, USA). The Scientific Committee included J.-P. Bourguignon (IHES), S.-Y.A. Chang (Princeton), C. LeBrun (Stony Brook), R. Schoen (Stanford), P. Yang (Princeton). The program was highlighted by a series of research conferences and "mini-courses"; i.e,, lecture series aimed at students and recent PhD recipients, in order to provide them with the necessary mathematical background to understand recent developments in the field(s). The conferences were: * Introductory Conference: Conformal and Kahler Geometry, September 17 –- 21, 2012; * Workshop: Geometry and Physics, October 08 -– 12, 2012; * Workshop: Geometric PDEs, November 5 -- 09, 2012; * Workshop: Recent Developments in Kahler Geometry, December 10 -- 14, 2012. The mini-courses for the program were: ``An introduction to conformal geometry'' by Olivier Druet (ENS-Lyon), September 10 -– 14, 2012; ``An introduction to the Tian-Yau-Donalsdon conjecture'' by Sean Paul (University of Wisconsin), September 10 -– 14, 2012; ``Introduction to the Cauchy problem in general relativity" by Piotr Chru?ciel'' (University of Vienna), October 8 –- 12; \ ``Min-max theory and the Willmore conjecture'' by Fernando Codá Marques (Instituto Nacional de Matemática Pura e Aplicada), October 15 –- 19, 2012; ``Asymptotic geometry of Einstein metrics'' by Olivier Biquard (Université Pierre et Marie Curie), October 15 -- 19, 2012; ``The Sphere Theorem'' by Simon Brendle (Stanford University), December 5--7, 2012; '`Curvature functionals, Kähler metrics, and the geometry of 4-manifolds'' by Claude LeBrun (Stony Brook University), December 3 -- 7, 2012. In all, the conferences featured speakers from sixteen different countries, while the NSF award supported the travel and lodging for over twenty participants from the U.S., mostly young researchers or PhD students.
