Category: Morse theory

Stratified Morse theory
In mathematics, stratified Morse theory is an analogue to Morse theory for general stratified spaces, originally developed by Mark Goresky and Robert MacPherson. The main point of the theory is to con
Novikov ring
In mathematics, given an additive subgroup , the Novikov ring of is the subring of consisting of formal sums such that and . The notion was introduced by Sergei Novikov in the papers that initiated th
Relative contact homology
In mathematics, in the area of symplectic topology, relative contact homology is an invariant of spaces together with a chosen subspace. Namely, it is associated to a contact manifold and one of its L
Perverse sheaf
The mathematical term perverse sheaves refers to a certain abelian category associated to a topological space X, which may be a real or complex manifold, or a more general topologically stratified spa
Morse theory
In mathematics, specifically in differential topology, Morse theory enables one to analyze the topology of a manifold by studying differentiable functions on that manifold. According to the basic insi
Taut submanifold
In mathematics, a (compact) taut submanifold N of a space form M is a compact submanifold with the property that for every the distance function is a perfect Morse function. If N is not compact, one n
Torus action
In algebraic geometry, a torus action on an algebraic variety is a group action of an algebraic torus on the variety. A variety equipped with an action of a torus T is called a T-variety. In different
Lusternik–Schnirelmann category
In mathematics, the Lyusternik–Schnirelmann category (or, Lusternik–Schnirelmann category, LS-category) of a topological space is the homotopy invariant defined to be the smallest integer number such
Gradient-like vector field
In differential topology, a mathematical discipline, and more specifically in Morse theory, a gradient-like vector field is a generalization of gradient vector field. The primary motivation is as a te
Morse homology
In mathematics, specifically in the field of differential topology, Morse homology is a homology theory defined for any smooth manifold. It is constructed using the smooth structure and an auxiliary m
Floer homology
In mathematics, Floer homology is a tool for studying symplectic geometry and low-dimensional topology. Floer homology is a novel invariant that arises as an infinite-dimensional analogue of finite-di
Jacobi set
In Morse theory, a mathematical discipline, Jacobi sets provide a method of studying the relationship between two or more Morse functions. For two Morse functions, the Jacobi set is defined as the set
Discrete Morse theory
Discrete Morse theory is a combinatorial adaptation of Morse theory developed by . The theory has various practical applications in diverse fields of applied mathematics and computer science, such as
Continuation map
In differential topology, given a family of Morse-Smale functions on a smooth manifold X parameterized by a closed interval I, one can construct a vector field on X × I whose critical points occur onl
Hessian matrix
In mathematics, the Hessian matrix or Hessian is a square matrix of second-order partial derivatives of a scalar-valued function, or scalar field. It describes the local curvature of a function of man
Lefschetz hyperplane theorem
In mathematics, specifically in algebraic geometry and algebraic topology, the Lefschetz hyperplane theorem is a precise statement of certain relations between the shape of an algebraic variety and th
Circle-valued Morse theory
In mathematics, circle-valued Morse theory studies the topology of a smooth manifold by analyzing the critical points of smooth maps from the manifold to the circle, in the framework of Morse homology