In symplectic geometry, the spectral invariants are invariants defined for the group of Hamiltonian diffeomorphisms of a symplectic manifold, which is closed related to Floer theory and Hofer geometry.
Vladimir Arnold conjectured that the number of fixed points of a generic Hamiltonian diffeomorphism of a compact symplectic manifold (M, ω) should be bounded from below by some topological constant of M, which is analogous to the Morse inequality.
Floer's definition adopted Witten's point of view on Morse theory.
Therefore, for any "good" Hamiltonian path Ht, a homology class α of M can be represented by a cycle in the Floer chain complex, formally a linear combination where ai are coefficients in some ring and xi are fixed points of the corresponding Hamiltonian diffeomorphism.
Formally, the spectral invariants can be defined by the min-max value Here the maximum is taken over all the values of the action functional AH on the fixed points appeared in the linear combination of αH, and the minimum is taken over all Floer cycles that represent the class α.