Symplectic Geometry & Floer Homology
Symplectic geometry studies the geometry of phase spaces, providing the mathematical framework for classical mechanics, while Floer homology revolutionizes our understanding through infinite-dimensional Morse theory, connecting to mirror symmetry and quantum field theory.
Fundamental Structures
Symplectic Manifold
A symplectic manifold is a pair $(M, \omega)$ where:
- $M$ is a smooth manifold of even dimension $2n$
- $\omega$ is a closed non-degenerate 2-form (symplectic form)
- $d\omega = 0$ (closed) and $\omega^n \neq 0$ (non-degenerate)
Darboux's Theorem
Every symplectic manifold is locally symplectomorphic to $(\mathbb{R}^{2n}, \omega_0)$ where:
$$\omega_0 = \sum_{i=1}^n dq_i \wedge dp_i$$This means symplectic geometry has no local invariants!
Phase Space Dynamics
Lagrangian Submanifolds
A submanifold $L \subset (M, \omega)$ is Lagrangian if:
- $\dim L = \frac{1}{2}\dim M = n$
- $\omega|_L = 0$ (isotropic condition)
Floer Homology
Floer Chain Complex
Critical points of action functional = periodic orbits
Floer Differential
Count pseudo-holomorphic strips between orbits
Arnold Conjecture (Proved via Floer Theory)
The number of fixed points of a Hamiltonian diffeomorphism on a compact symplectic manifold is at least the sum of the Betti numbers:
$$\#\text{Fix}(\phi_H^1) \geq \sum_i b_i(M)$$Pseudo-holomorphic Curves
J-holomorphic Maps
A map $u: (S, j) \to (M, J)$ is pseudo-holomorphic if:
$$du \circ j = J \circ du$$Mirror Symmetry
A duality between symplectic geometry of one space and complex geometry of another:
Fukaya category of $(X, \omega)$ ≅ Derived category of coherent sheaves on $\check{X}$
This deep connection has revolutionized both mathematics and string theory.
Fukaya Categories
The Fukaya category $\mathcal{F}uk(M, \omega)$ has:
- Objects: Lagrangian submanifolds with extra data
- Morphisms: Floer cochain complexes
- Composition: Counts of pseudo-holomorphic triangles
Moment Maps and Reduction
For a Hamiltonian $G$-action on $(M, \omega)$, the moment map $\mu: M \to \mathfrak{g}^*$ satisfies:
$$d\langle \mu, \xi \rangle = \iota_{\xi_M} \omega$$Gromov-Witten Invariants
Counting Curves
Gromov-Witten invariants count pseudo-holomorphic curves in symplectic manifolds:
$$\langle \alpha_1, ..., \alpha_n \rangle_{g,n,A} = \int_{[\overline{\mathcal{M}}_{g,n}(X,A)]^{vir}} \prod_{i=1}^n ev_i^*(\alpha_i)$$Classical Examples
Example: Cotangent Bundle
$T^*Q$ with canonical symplectic form $\omega = \sum dp_i \wedge dq_i$
Zero section is Lagrangian, generating function gives Lagrangians
Example: Complex Projective Space
$\mathbb{CP}^n$ with Fubini-Study form is symplectic
Real projective space $\mathbb{RP}^n \subset \mathbb{CP}^n$ is Lagrangian
Foundational Papers
The paper that started Floer homology revolution.
Introduction of J-holomorphic curves, revolutionizing symplectic topology.
The foundational conjecture connecting symplectic and algebraic geometry.
Rigorous foundation for quantum cohomology and Gromov-Witten theory.
Revolutionary Applications
Classical Mechanics
Symplectic geometry is the natural language for Hamiltonian mechanics and geometric quantization.
String Theory
Mirror symmetry originated in physics, connecting different Calabi-Yau compactifications.
Low-dimensional Topology
Heegaard Floer homology revolutionized 3 and 4-manifold topology.
Leading Research Centers
- Stanford - Yakov Eliashberg's group
- MIT - Denis Auroux, Paul Seidel
- Columbia - Mohammed Abouzaid
- IAS Princeton - Helmut Hofer
- ETH Zürich - Symplectic geometry group