Symplectic Geometry, Poisson Geometry, and Beyond

TL;DR

This paper introduces the fundamental concepts and techniques of symplectic and Poisson geometry, highlighting their origins in classical mechanics and their applications in Lie theory and mathematical physics.

Contribution

It provides an accessible overview of key objects and methods in symplectic and Poisson geometry, connecting classical physics examples to modern mathematical applications.

Findings

Introduction of essential symplectic and Poisson objects

Connection of geometric structures to classical physics

Applications to Lie theory and mathematical physics

Abstract

Symplectic and Poisson geometry emerged as a tool to understand the mathematical structure behind classical mechanics. However, due to its huge development over the past century, it has become an independent field of research in differential geometry. In this lecture notes, we will introduce the essential objects and techniques in symplectic geometry (e.g Darboux coordinates, Lagrangian submanifolds, cotangent bundles) and Poisson geometry (e.g symplectic foliations, some examples of Poisson structures). This geometric approach will be motivated by examples from classical physics, and at the end we will explore applications of symplectic and Poisson geometry to Lie theory and other fields of mathematical physics.