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.
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 to higher symplectic geometry. This involves multisymplectic geometry and the geometry of symplectic Lie n-algebroids. And their combination.
|symplectic Lie n-algebroid||Lie integrated smooth ∞-groupoid = moduli ∞-stack of fields of -d sigma-model||higher symplectic geometry||d sigma-model||dg-Lagrangian submanifold/ real polarization leaf||= brane||(n+1)-module of quantum states in codimension||discussed in:|
|0||symplectic manifold||symplectic manifold||symplectic geometry||Lagrangian submanifold||–||ordinary space of states (in geometric quantization)||geometric quantization|
|1||Poisson Lie algebroid||symplectic groupoid||2-plectic geometry||Poisson sigma-model||coisotropic submanifold (of underlying Poisson manifold)||brane of Poisson sigma-model||2-module = category of modules over strict deformation quantiized algebra of observables||extended geometric quantization of 2d Chern-Simons theory|
|2||Courant Lie 2-algebroid||symplectic 2-groupoid||3-plectic geometry||Courant sigma-model||Dirac structure||D-brane in type II geometry|
|symplectic Lie n-algebroid||symplectic n-groupoid||(n+1)-plectic geometry||AKSZ sigma-model|
(adapted from Ševera 00)
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.
J.J. Duistermaat, Fourier integral operators, Progress in Mathematics, Birkhäuser 1995 (and many other references at microlocal analysis).
N. R. Wallach, Symplectic geometry and Fourier analysis, Math. Sci. Press, Brookline, Mass., 1977.