Symplectic, Poisson, and Noncommutative Geometry

Symplectic geometry originated in physics, but it has flourished as an independent subject in mathematics, together with its offspring, symplectic topology. Symplectic methods have even been applied back to mathematical physics. Noncommutative geometry has developed an alternative mathematical quantization scheme based on a geometric approach to operator algebras. Deformation quantization, a blend of symplectic methods and noncommutative geometry, approaches quantum mechanics from a more algebraic viewpoint, as it addresses quantization as a deformation of Poisson structures. This volume contains seven chapters based on lectures given by invited speakers at two May 2010 workshops held at the Mathematical Sciences Research Institute: Symplectic and Poisson Geometry in Interaction with Analysis, Algebra and Topology (honoring Alan Weinstein, one of the key figures in the field) and Symplectic Geometry, Noncommutative Geometry and Physics. The chapters include presentations of previously unpublished results and comprehensive reviews, including recent developments in these areas.1. Flexible Weinstein manifolds Kai Cieliebak and Yakov Eliashberg; 2. The Hirzebruch-Riemann-Roch theorem in true genus-0 quantum K-theory Alexander Givental and Valentin Tonita; 3. On some deformations of Fukaya categories Hiroshige Kajiura; 4. Morphisms of CohFT algebras and quantization of the Kirwan map K. L. Nguyen, Chris Woodward and Fabian Ziltener; 5. Deformation of expressions for elements of an algebra Hideki Omori, Yoshiaki Maeda and Naoya Miyazaki; 6. Microlocal theory of sheaves in symplectic topology Pierre Schapire; 7. Algebra + homotopy = operad Bruno Vallette.