Morse theory

Morse theory relates a smooth function’s critical points and their local shapes to the topology of the manifold on which the function is defined.

Morse function: A smooth function whose critical points are all nondegenerate. Morse theory extracts topological information from functions with this generic, well-behaved critical-point structure.
Smooth manifold: A space that locally resembles Euclidean space and has smoothly compatible coordinate charts. Morse theory studies smooth functions defined on these spaces.
Morse inequalities: Inequalities relating the numbers of critical points of each index to a manifold’s Betti numbers. They turn a function’s critical-point data into constraints on the manifold’s homology.
Marston Morse: An American mathematician who developed the theory now called Morse theory. His work established how critical points of smooth functions encode topological structure.
Morse–Smale system: A dynamical system whose stable and unstable manifolds intersect transversely and whose nonwandering set is simple. Transversality strengthens gradient-flow behavior used in refined Morse-theoretic constructions.
Hessian matrix: The matrix of second partial derivatives of a scalar function. At a critical point, its eigenvalues determine nondegeneracy and the point’s Morse index.
Critical point: A point where a differentiable function’s derivative vanishes or is not defined. The locations of these points are the basic data Morse theory turns into topology.
Morse–Bott theory: A generalization of Morse theory allowing critical sets that are smooth submanifolds. It handles functions whose critical points occur in families rather than as isolated points.
Calculus of variations: The mathematical study of extrema of functionals over spaces of functions or paths. Morse developed foundational methods while studying critical points in variational problems.
Singularity theory: The study of local forms and classifications of singularities of maps and functions. It addresses degenerate critical points that ordinary Morse theory excludes.

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