Symplectic Geometry

Explore the fundamental theory of symplectic manifolds through this comprehensive mathematics course that examines the crucial role symplectic structures play in modern mathematics and physics. Master the basic local theory including the essential Darboux theorem, which provides the foundation for understanding symplectic coordinates. Investigate the deep connections between symplectic geometry and complex geometry, particularly Kähler geometry, and discover how these relationships illuminate both fields. Delve into Hamiltonian mechanics to understand how symplectic geometry provides the natural mathematical framework for classical mechanics and dynamical systems. Study moment maps and symplectic reduction techniques, powerful tools for analyzing symmetries in symplectic manifolds and reducing complex systems to simpler ones. Examine advanced topics including toric varieties and their symplectic properties, hyperkähler structures that combine symplectic and complex geometry, quantization procedures that bridge classical and quantum mechanics, Fukaya categories from algebraic topology, and the profound connections to mirror symmetry. Progress through 24 structured classes that build systematically from foundational concepts to cutting-edge research areas, gaining both theoretical understanding and practical techniques for working with symplectic manifolds in various mathematical and physical contexts.