Singularity theory

Singularity theory studies points where maps, spaces, or solutions fail to be smooth or regular. It classifies these failures and examines how they change under perturbation.

Differential map: A function between spaces equipped with a notion of differentiability. Singularities commonly arise where a differentiable map loses its usual local behavior.
Map germ: The behavior of a map restricted to an arbitrarily small neighborhood of a point. Singularity theory studies local behavior, discarding distant parts of a map.
Fold singularity: A map singularity locally equivalent to a projection with one coordinate squared. The fold is the simplest stable singularity of many smooth maps.
Hassler Whitney: An American mathematician who founded major parts of differential topology and geometric singularity theory. Whitney classified stable singularities of smooth maps and developed stratification ideas.
Caustic (optics): A bright curve or surface formed where neighboring light rays concentrate. Caustics are visible images of singularities in projections of ray families.
Jacobian matrix: The matrix of first derivatives of a differentiable map at a point. Its rank detects many singular points of maps between smooth spaces.
Normal form: A simplified representative that captures an object up to a chosen equivalence. Normal forms expose the essential structure of a singularity.
Cusp singularity: A singularity with a cusp-shaped local model, typically appearing where folds meet. The cusp illustrates how a more complex stable singularity organizes nearby folds.
René Thom: A French mathematician known for cobordism theory and foundational work on singularities and catastrophe theory. Thom developed structural stability and influential classification methods for singularities.
Lagrangian submanifold: A submanifold of a symplectic manifold on which the symplectic form vanishes and whose dimension is half the ambient dimension. Projections of Lagrangian submanifolds generate singularities in optics and mechanics.

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