Symplectic geometry is a branch of differential geometry studying symplectic manifolds and some generalizations; it originated as a formalization of the mathematical aparatus of classical mechanics and geometric optics (and the related WKB-method in quantum mechanics and, more generally, the method of stationary phase in harmonic analysis?). A wider branch including symplectic geometry is Poisson geometry? and a sister branch in odd dimensions is contact geometry?. A special and central role in the subject belongs to certain real-like half-dimensional submanifolds, called lagrangian (or Lagrangean) submanifolds?, which are in some sense classical points. Symplectic geometry radically changed after the 1985 article of Gromov on pseudoholomorphic curves and the subsequent work of Floer giving birth to symplectic topology or “hard methods” of symplectic geometry.
In its application to physics, symplectic geometry is the fundamental mathematical language for Hamiltonian mechanics, geometric quantization, geometrical optics.
A tremendous amount of insight into higher Lie theory (Lie groupoids, Lie ∞-groupoids, Lie ∞-algebroids) has derived from Alan Weinstein’s long-term project of understanding the role of symplectic geometry in geometric quantization. See there for more details.
Zoran Škoda: it is true that large influence of Weinstein’s program can not be overestimated, but it is not the origin of these considerations; it rather builds up on earlier fundamental works of Kirillov, Kostant, Souriau who invented geometric quantization, all of them originally in symplectic context; and the florishing of the subject from mid 1960s till mid 1980-s is related to their work; and other related tracks of Guillemin, Sternberg, Kashiwara, Karasev, Arnold and so on; and vast developments in harmonic analysis and representation theory (Kostant, Auslander, Vogan, Wallach, Stein…), microlocal analysis (Kashiwara, Saito, Hormander, Maslov, Karasev, Duistermaat…), integrable systems/quantum groups (this is more into more general Poisson geometry: Lie-Poisson groups, classical r-matrices, bihamiltonian systems…), and related approaches to quantization (Berezin method, coherent states…).
There is a vertical categorification of symplectic geometry, called multisymplectic geometry . For more on this see
T-duality and mirror symmetry interchange the symplectic data and geometric structure data. Some cases of it can be unified in a broader concept of generalized complex geometry.
The notion of symplectic geometry may be understood as the mathematical structure that underlies the physics of Hamiltonian mechanics. A classical monograph that emphasizes this point of view is
For more on this see Hamiltonian mechanics. Symplectic geometry is also involved in geometric optics? and the study of oscillatory integrals and microlocal analysis. For a book concentrating on these topics which is free online see
Here is a telegraphic version of the story:
Conservation lawas arising from symmetries have been formalized as moment maps by Kirillov, Kostant and Souriau in late 1960s. Elimination of nodes procedure has been made rigorous by Marsden and Weinstein and, independently, by Meyer, as symplectic reduction? (symplectic quotient construction).
In the early 1980s Mumford observed that symplectic quotients are closely related to geometric invariant theory? quotients and that many moduli spaces important in algebraic geometry and in mathematical physics can be realized as symplectic quotient?s. Atiyah and Bott used this point of view in “The moment map and equivariant cohomology” to construct cohomology classes of moduli spaces of flat connections on Riemann surfaces. In “Two dimensional gauge theories revisited” Witten conjectured a method for computing the intersection pairing?s of cohomology classes of symplectic quotients. The work of Atiyah and Bott and Witten’s conjecture stimulated a large research effort to understand the topology of symplectic quotients in terms of the equivariant cohomology of the original spaces. Witten’s conjecture was proved by Jeffrey and Kirwan several years later.