The Symplectic Geometry
Symplectic geometry is the geometry of phase space. It was born from Hamiltonian mechanics, but it has become an autonomous field of mathematics with its own questions, its own theorems, and its own culture. It studies even-dimensional manifolds equipped with a closed, non-degenerate 2-form — the symplectic form — and the transformations that preserve it.
The subject sits at an unusual intersection. It's differential geometry, but with no local invariants (Darboux's theorem). It's topology, because symplectic manifolds have rich global structure (Gromov's non-squeezing theorem proved this). It's algebra, through Poisson algebras and deformation quantization. And it's analysis, through Floer theory and pseudoholomorphic curves. This multiplicity of connections is both the strength and the difficulty of the field.
The basic objects are symplectic manifolds (M, omega). A symplectic manifold is a smooth manifold M of even dimension 2n with a 2-form omega that is closed (d omega = 0) and non-degenerate (omega^n is a volume form). The simplest example is R^{2n} with omega_0 = sum dq^i wedge dp_i. Darboux's theorem says every symplectic manifold is locally symplectomorphic to (R^{2n}, omega_0). There is no symplectic curvature. The geometry is entirely a question of global topology.
The Hamiltonian formalism is the bridge to physics. Given a function H on M, the Hamiltonian vector field X_H is defined by i_{X_H} omega = dH. The flow of X_H gives the time evolution. The Poisson bracket {f, g} = omega(X_f, X_g) encodes the algebra of observables. Conservation laws correspond to functions that Poisson-commute with H. This is the complete machinery of classical mechanics, reformulated in the language of symplectic geometry.
The quantization problem is one of the central motivations. Can one find a rule that associates to each function f on M an operator pi(f) on some Hilbert space, such that pi({f, g}) = (1/i\hbar)[pi(f), pi(g)]? Groenewold and van Vleck proved that no such rule exists that works for all functions. But geometric quantization, deformation quantization, and other approaches give partial solutions, and the relationship between symplectic geometry and quantum mechanics remains one of the most productive tensions in mathematical physics.
Gromov's non-squeezing theorem (1985) was a watershed. It showed that symplectic geometry does have non-trivial content — you cannot symplectically embed a ball of radius R into a cylinder of radius r if R > r, even though there's no local obstruction. This is a genuinely global phenomenon, and it launched symplectic topology as a field. The proof uses pseudoholomorphic curves, a technique invented by Gromov that has since become central in symplectic geometry.
Floer homology (1988) extended this further. Andreas Floer constructed an infinite-dimensional Morse theory for the space of loops in a symplectic manifold, using pseudoholomorphic cylinders in the cylinder R × M. The resulting homology groups are invariants of the symplectic manifold and have deep connections to the dynamics of Hamiltonian systems. The Arnold conjecture, about the number of periodic orbits of Hamiltonian vector fields, falls out of Floer theory.
The field continues to evolve. Mirror symmetry, originally a prediction from string theory, has led to deep theorems in symplectic geometry via the work of Kontsevich and others. The Fukaya category of a symplectic manifold encodes its Lagrangian submanifolds and their intersection theory. Homological mirror symmetry conjectures that this category is equivalent to the derived category of coherent sheaves on a "mirror" complex manifold. These connections continue to drive new developments.