Lectures on Symplectic Geometry

Abstract

ForewordThese notes approximately transcribe a 15-week course on symplectic geometry I taught at UC Berkeley in the Fall of 1997.The course at Berkeley was greatly inspired in content and style by Victor Guillemin, whose masterly teaching of beautiful courses on topics related to symplectic geometry at MIT, I was lucky enough to experience as a graduate student. I am very thankful to him! That course also borrowed from the 1997 Park City summer courses on symplectic geometry and topology, and from many talks and discussions of the symplectic geometry group at MIT. Among the regular participants in the MIT informal symplectic seminar 93-96, I would References 199 Index 207 IntroductionThe goal of these notes is to provide a fast introduction to symplectic geometry. A symplectic form is a closed nondegenerate 2-form. A symplectic manifold is a manifold equipped with a symplectic form. Symplectic geometry is the geometry of symplectic manifolds. Symplectic manifolds are necessarily even-dimensional and orientable, since nondegeneracy says that the top exterior power of a symplectic form is a volume form. The closedness condition is a natural differential equation, which forces all symplectic manifolds to being locally indistinguishable. (These assertions will be explained in Lecture 1 and Homework 2.)The list of questions on symplectic forms begins with those of existence and uniqueness on a given manifold. For specific symplectic manifolds, one would like to understand the geometry and the topology of special submanifolds, the dynamics of certain vector fields or systems of differential equations, the symmetries and extra structure, etc.Two centuries ago, symplectic geometry provided a language for classical mechanics. Through its recent huge development, it conquered an independent and rich territory, as a central branch of differential geometry and topology. To mention just a few key landmarks, one may say that symplectic geometry began to take its modern shape with the formulation of the Arnold conjectures in the 60's and with the foundational work of Weinstein in the 70's. A paper of Gromov [49] in the 80's gave the subject a whole new set of tools: pseudo-holomorphic curves. Gromov also first showed that important results from complex Kähler geometry remain true in the more general symplectic category, and this direction was continued rather dramatically in the 90's in the work of Donaldson on the topology of symplectic manifolds and their symplectic submanifolds, and in the work of Taubes in the context of the Seiberg-Witten invariants. Symplectic geometry is significantly stimulated by important interactions with global analysis, mathematical physics, low-dimensional topology, dynamical systems, algebraic geometry, integrable systems, microlocal analysis, partial differential equations, representation theory, quantization, equivariant cohomology, geometric combinatorics, etc.As a curiosity, note that two centuries ago the name symplectic geometry did not exist. If you consult a major English dictionary, ...

N $$ \mathcal{N} $$ =2 supersymmetric field theories on 3-manifolds with A-type boundaries

“…The last condition is called ω-compatibility [40]. In a local description with coordinates (φ a ,φc) the metric is represented as 18) and the two-form ω is represented as ω αc ∝ G ac dφ a ∧dφc.…”

Catalytic Quantum Error Correction

“…We need to show that the conditions from item 1 are satisfied for the next round (i := i + 1). Condition (a) holds because {u i , v i , w (9) and (10). Regarding condition (c), if m ′ = 0 then it holds because U and V did not change from the previous round.…”

A Method for Choosing an Initial Time Eigenstate in Classical and Quantum Systems

“…The vectors X F and X G form a basis of a symplectic vector space of dimension two, under the skew-symmetric map Ω(u, v) := {u, v}, with rank two [1]. Under the standard symplectic form Ω(z, z ′ ) = p · r ′ − p ′ · r, the vector fields X F and X G comply with the fundamental relation, with different signs:…”