Seminar: Singular Symplectic Manifolds and Geometric Quantization

Abstract: Singular symplectic structures, a large class of Poisson structures, have been studied in the past decade through the lenses of quantization, Delzant-type classification, and Hamiltonian Dynamics. Most recently, Lin, Loizides, Sjamaar, and Song give a geometric quantization of the closed log-symplectic manifolds, where the symplectic form has poles of order 1 along hypersurfaces with normal crossings. This quantization is an index of a special Spin^c Dirac operator. This quantization in the Hamiltonian setting commutes with reduction. In this work, we extend this result to $b^k$-symplectic manifolds. This time the poles are of order k. We will explain a quantization approach that exploits the integrability of certain algebroids, thereby extending quantization to larger classes of singular symplectic manifolds, such as E-symplectic manifolds.

Host: Yanli Song