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| Floer theory | |
|---|---|
| Name | Floer theory |
| Field | Mathematics |
| Subfield | Symplectic geometry, Low-dimensional topology |
| Introduced | 1980s |
| Founder | Andreas Floer |
| Notable contributors | Andreas Floer; Yakov Eliashberg; Paul Seidel; Maxim Kontsevich; Kenji Fukaya; Mohammed Abouzaid; Ivan Smith; Denis Auroux; Katrin Wehrheim; Chris Woodward; Helmut Hofer; Dusa McDuff; Michael Hutchings |
Floer theory Floer theory is a collection of techniques and invariants in Symplectic geometry and Low-dimensional topology that use infinite-dimensional variational methods to extract algebraic information from analytical moduli problems. Developed in the 1980s, it connects ideas from Morse theory, Partial differential equations, and categorical frameworks found in Homological mirror symmetry. Floer-theoretic constructions produce homology groups, functors, and spectral invariants applied across Hamiltonian dynamics, knot theory, and gauge theory.
Floer-theoretic approaches start by associating to geometric data—such as a pair of Lagrangian submanifolds, a Hamiltonian function on a symplectic manifold, or a three-manifold with a Spinc structure—an infinite-dimensional manifold of maps whose critical points correspond to geometric intersections or closed trajectories. Drawing upon analogies with Marston Morse and the gradient flow equations used by Morse theory, Floer introduced analytic tools resembling the study of elliptic operators seen in Atiyah–Singer index theorem contexts. Subsequent work by figures like Andreas Floer, Helmut Hofer, Paul Seidel, Kenji Fukaya, and Maxim Kontsevich established algebraic and categorical frameworks linking to the Fukaya category, the Seiberg–Witten invariant, and structures appearing in Donaldson theory.
The origins trace to Andreas Floer’s foundational contributions inspired by problems in Arnold conjecture and periodic orbits in Hamiltonian dynamics. Early milestones include Floer’s solutions to variational problems informed by the analytic techniques of S. Smale and the index-theoretic insights of Michael Atiyah and Isadore Singer. Parallel developments in Gauge theory by Simon Donaldson and later by Edward Witten via Donaldson–Thomas theory influenced the growth of instanton and monopole Floer homologies. Influential events such as workshops at Institute for Advanced Study and collaborations among researchers at Princeton University, University of Cambridge, and IHÉS accelerated formalization, while institutions like Clay Mathematics Institute funded related research programs.
Analytical foundations rely on elliptic regularity, transversality, compactness theorems, and gluing constructions familiar from Partial differential equations and index theory. Central analytical objects include the Cauchy–Riemann type equations studied by Gromov in pseudoholomorphic curve theory, the nonlinear Fredholm theory cultivated by Hofer–Salamon and others, and the compactness framework developed in studies of bubbling phenomena related to Gromov compactness. Technical tools incorporate virtual perturbation machinery introduced in projects involving Fukaya–Oh–Ohta–Ono and polyfold theory from Hofer–Wysocki–Zehnder, as well as obstruction bundle techniques used by contributors like Katrin Wehrheim and Dusa McDuff.
Variants of Floer constructions reflect the geometric context. Hamiltonian Floer homology connects to the Arnold conjecture and yields spectral invariants via work by Yakov Eliashberg and Jean-Claude Sikorav. Lagrangian Floer homology underpins the definition of the Fukaya category developed by Kenji Fukaya, Paul Seidel, and collaborators. Instanton Floer homology originates from Donaldson theory and was extended in the monopole context by Clifford Taubes and Peter Kronheimer with Tomi Mrowka. Embedded contact homology, introduced by Michael Hutchings, relates to Seiberg–Witten theory proven by links established by Clifford Taubes. Variants include symplectic field theory advanced by Eliashberg–Givental–Hofer, Heegaard Floer homology by Peter Ozsváth and Zoltán Szabó, and quilted Floer theory by Wehrheim–Woodward.
Applications span proofs and obstructions: verifying instances of the Arnold conjecture, distinguishing exotic symplectic structures studied at McDuff–Polterovich seminars, and detecting nondisplaceable Lagrangians after research by Polterovich and Lalonde. In low-dimensional topology, Heegaard Floer homology produced knot and three-manifold invariants instrumental in resolving conjectures related to fibred knots and contact structures investigated by John Etnyre and Ko Honda. Instanton and monopole Floer homologies inform results in four-manifold topology rooted in Donaldson invariants and Seiberg–Witten invariants, with computational input from researchers at Harvard University, Princeton University, and ETH Zurich.
Floer ideas are central in formulations of Homological mirror symmetry conjectured by Maxim Kontsevich, relating the Fukaya category of a symplectic manifold to the derived category of coherent sheaves on a mirror complex manifold featured in work by Dmitry Orlov and Paul Seidel. Developments in A∞-categories, pioneered by Jim Stasheff and incorporated by Kenji Fukaya and Bernhard Keller, formalize algebraic structures arising from Floer complexes. Categorical equivalences with consequences for deformation quantization and enumerative predictions engage researchers affiliated with IHÉS, Institut des Hautes Études Scientifiques, and university groups including Oxford University and MIT.
Concrete computations occur for toric varieties analyzed by Denis Auroux and for symplectic surfaces studied by Ivan Smith. Knot invariants from Heegaard Floer theory yield concordance invariants applied by Tim Cochran and Peter Ozsváth. Spectral invariants and capacity-type quantities link to work by Leonid Polterovich and Claude Viterbo. Enumerative predictions, counts of pseudoholomorphic curves, and wall-crossing phenomena draw on contributions from Maxim Kontsevich and Y.-G. Oh. Active computational programs continue in research groups at Caltech, University of California, Berkeley, and University of Toronto refining algorithms for Floer-type invariants.