Advances in Geometry, 1999 Volume 1 Progress in Mathematics Series, Vol. 172

This book is an outgrowth of the activities of the Center for Geometry and Mathematical Physics (CGMP) at Penn State from 1996 to 1998. The Center was created in the Mathematics Department at Penn State in the fall of 1996 for the purpose of promoting and supporting the activities of researchers and students in and around geometry and physics at the university. The CGMP brings many visitors to Penn State and has ties with other research groups; it organizes weekly seminars as well as annual workshops The book contains 17 contributed articles on current research topics in a variety of fields: symplectic geometry, quantization, quantum groups, algebraic geometry, algebraic groups and invariant theory, and character­ istic classes. Most of the 20 authors have talked at Penn State about their research. Their articles present new results or discuss interesting perspec­ tives on recent work. All the articles have been refereed in the regular fashion of excellent scientific journals. Symplectic geometry, quantization and quantum groups is one main theme of the book. Several authors study deformation quantization. As­ tashkevich generalizes Karabegov's deformation quantization of Kahler manifolds to symplectic manifolds admitting two transverse polarizations, and studies the moment map in the case of semisimple coadjoint orbits. Bieliavsky constructs an explicit star-product on holonomy reducible sym­ metric coadjoint orbits of a simple Lie group, and he shows how to con­ struct a star-representation which has interesting holomorphic properties.

On Karabegov’s Quantizations of Semisimple Coadjoint Orbits.- Exotic Differential Operators on Complex Minimal Nilpotent Orbits.- The Geometry Surrounding the Arnold-Liouville Theorem.- Symmetric Spaces and Star Representations.- Hyperplane Arrangements, Springer Representations and Exponents.- Comparison of the Beilinson-Chern Classes with the Chern-Cheeger-Simons Classes.- Geometric Construction of Quillen Line Bundles.- Quadratic Algebras, Dunkl Elements, and Schubert Calculus.- Logarithmic Forms with Twisted Coefficients.- Higher Holonomies, Geometric Loop Groups and Smooth Deligne Cohomology.- Moduli Spaces of Linkages and Arrangements.- Moduli Spaces of Flat Connections on 2-Manifolds, Cobordism, and Witten’s Volume Formulas.- A Rigidity Property for QuantumSU(3) Groups.- On the Cohomology Ring of an Algebra.- On A Quantum Version of Pieri’s Formula.- Some Non-Koszul Algebras.- A Variety of Solutions to the Yang-Baxter Equation.