symplectic geometry



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.


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…).



(adapted from Ševera 00)


Introductions include

  • Rolf Berndt, An introduction to symplectic geometry (pdf)

Discussion from the point of view of homological algebra of abelian sheaves is in

  • C. Viterbo, An introduction to symplectic topology through sheaf theory (2010) (pdf)

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, geometric quantization and the study of oscillatory integrals and microlocal analysis. Books concentrating on these topics include

Revised on September 19, 2013 00:58:43 by Urs Schreiber (