An introduction to symplectic geometry

Více o knize

Symplectic geometry is a vital area of contemporary mathematical research, essential for understanding dynamical systems, differential equations, algebraic geometry, topology, mathematical physics, and Lie group representations. This book serves as an accessible introduction to symplectic geometry, requiring only a general background in analysis and linear algebra. It begins with the fundamentals of symplectic vector spaces, followed by an exploration of symplectic manifolds. Key classical results, such as Darboux's theorem, are presented alongside modern concepts like symplectic capacity and pseudoholomorphic curves, which have transformed the field. The text covers major examples of symplectic manifolds, including the cotangent bundle, Kähler manifolds, and coadjoint orbits. Important ideas like Hamiltonian vector fields and the Poisson bracket are examined, along with their connections to contact manifolds. The author discusses the relationship between symplectic geometry and mathematical physics, particularly focusing on the moment map and symplectic reduction, which are crucial for simplifying physical systems and generating new symplectic manifolds. The final chapter addresses quantization, linking classical and quantum mechanics through symplectic methods. Several appendices offer additional material on vector bundles, cohomology, and Lie groups. This clear and concise presentation makes it an excellent resource for gra