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Chin-Yu Hsiao (Academia Sinica)
2024-07-19  15:00 - 16:00
Room 202, Astronomy and Mathematics Building

The famous geometric quantization theorem of Guillemin and Sternberg  states that for a compact pre-quantizable symplectic manifold admitting a Hamiltonian action of a compact  Lie group, the principle of "quantization commutes with reduction" holds. When a complex manifold has singularities or momentum map is not regular, the study of "quantization commutes with reduction" principle plays an important role in modern symplectic geometry, complex algebraic geometry and mathematical physics. According to J. Burning, "boundary" is also some kind of singularities. In this work, we establish geometric quantization for CR manifolds (boundary version of classical of Guillemin and Sternberg  geometric quantization ) and complex manifolds with boundary. An important difference between the CR setting and the K\"ahler/symplectic setting is that the quantum spaces in the case of a compact K\"ahler/symplectic manifolds are finite dimensional, whereas for The CR manifolds that we consider the quantum spaces are infinite dimensional. For this purpose, we need new idea,  we develop a $G$-invariant Fourier integral operator calculus to study  geometric quantization.