Singularity (mathematics)

Singularity (mathematics)
Name	Singularity (mathematics)
Field	Mathematics
Key concepts	Complex analysis, Algebraic geometry, Differential topology, Singularity theory

Singularity (mathematics) A mathematical singularity denotes a point, locus, or parameter value where an otherwise regular mathematical object fails to satisfy the standard properties expected in contexts studied by Augustin-Louis Cauchy, Carl Friedrich Gauss, Bernhard Riemann, Henri Poincaré, Sofia Kovalevskaya. Singularities arise across Complex analysis, Algebraic geometry, Differential topology, Dynamical systems, Functional analysis and connect to work by Alexander Grothendieck, René Thom, John Milnor, William Thurston, Michael Atiyah.

Definition and Types

A singularity is typically defined as a point where a function, manifold, map, or algebraic variety ceases to be well-behaved in the sense used by Augustin-Louis Cauchy, Bernhard Riemann, Oscar Zariski, Alexander Grothendieck. In Complex analysis this means a point where a holomorphic function fails to be analytic as in the theory of Riemann sphere and Laurent series developed by Joseph-Louis Lagrange, Carl Gustav Jacob Jacobi, Karl Weierstrass. In Algebraic geometry singularities are points where a variety is not a smooth manifold in the sense of Élie Cartan or where the Jacobian criterion of David Hilbert and Emmy Noether fails. In Differential topology singularities of maps are studied using transversality theorems of René Thom and Stephen Smale, and in Singularity theory catastrophes relate to classifications by Vladimir Arnol'd and John Nash.

Isolated and Non-isolated Singularities

An isolated singularity, as in examples studied by Bernhard Riemann and Augustin-Louis Cauchy, is a singular point with a punctured neighborhood free of other singular points; important instances occur on the Riemann sphere and in the study of meromorphic functions following Karl Weierstrass. Non-isolated singularities occur in algebraic varieties such as nodal curves and surface singularities analyzed by Oscar Zariski, Kunihiko Kodaira, Heisuke Hironaka. Families of singularities parameterized by moduli spaces studied by David Mumford and Pierre Deligne exhibit loci of accumulation and require techniques from Mumford–Tate group theory and deformation theory developed by Grothendieck.

Classification by Behavior (Removable, Pole, Essential)

In Complex analysis isolated singularities admit the classical trichotomy: removable singularities, poles, and essential singularities, a taxonomy tied to the Laurent series and the Casorati–Weierstrass theorem related to work by Carlo Emilio Bonferroni and Georg Cantor in function theory. Removable singularities appear in settings treated by Riemann mapping theorem methods of Bernhard Riemann; poles correspond to principal parts as in Cauchy residue theorem applications of Augustin-Louis Cauchy; essential singularities yield dense image behavior per Casorati–Weierstrass and Picard theorem of Émile Picard. Classification of singularities of analytic mappings also uses invariants introduced by John Milnor, Vladimir Arnol'd, René Thom and Gaston Darboux.

Singularities in Several Complex Variables

Several complex variables bring phenomena absent in one-variable theory, as illustrated by domains studied by Hermann Weyl, Kiyoshi Oka, Lars Ahlfors and Shoshichi Kobayashi. Hartogs' phenomenon, Bergman kernel methods of Stefan Bergman, and pseudoconvexity concepts linked to Hermann Reyer and Kiyoshi Oka show that isolated singularities behave differently on complex manifolds of dimension greater than one. Tools from sheaf cohomology developed by Jean-Pierre Serre and resolution techniques by Heisuke Hironaka and Alexander Grothendieck are central to understanding analytic and algebraic singular loci in higher dimensions.

Algebraic and Geometric Singularities

Algebraic singularities on varieties and schemes, as formalized by Alexander Grothendieck and Oscar Zariski, are points where the local ring fails to be regular in the sense of Krull dimension and Noetherian ring theory of Emmy Noether and David Hilbert. Geometric singularities studied by William Thurston, Michael Atiyah, Raoul Bott and Mikhail Gromov include cone points, cusps, nodes and orbifold singularities appearing in moduli problems treated by Pierre Deligne and David Mumford. Singularity invariants such as Milnor number, Tjurina number, and multiplicity connect to intersection theory of Jean-Pierre Serre and duality theorems by Alexander Grothendieck.

Resolution and Regularization Techniques

Resolution of singularities, proven in characteristic zero by Heisuke Hironaka, produces a smooth model via blowups and alterations inspired by techniques of Oscar Zariski and later refinements by Tetsuji Shioda and Jan Denef. Regularization methods include analytic continuation from Riemann, renormalization ideas related to work by Richard Feynman and algebraic desingularization via normalization, normalization algorithms of Bernard Teissier, and minimal model program contributions by Shigefumi Mori and Yakov Eliashberg. Other techniques involve Morse theory of Marston Morse, stratification and Whitney conditions developed by Hassler Whitney, and perverse sheaves of Alexander Beilinson and Joseph Bernstein.

Applications and Examples

Singularities play roles in the study of dynamical systems by Stephen Smale, bifurcation theory of René Thom, and catastrophe models of Vladimir Arnol'd. Examples include the node and cusp in plane curve singularities studied by Oscar Zariski and Federigo Enriques, ADE classifications linked to Lie groups of Élie Cartan and singularity types appearing in the work of Michael Atiyah and Isadore Singer on index theory. In mathematical physics, singularities appear in models considered by Paul Dirac, Albert Einstein, Satyendra Nath Bose, and in string theory contexts explored by Edward Witten and Cumrun Vafa, where Calabi–Yau degeneration and orbifold singularities influence dualities studied by Juan Maldacena. Computational methods for singularity detection and deformation draw on algorithms by David Cox, John Little, Donal O'Shea, and applications in robotics and control reference work by Richard Murray and Jean-Jacques Slotine.

Category:Complex analysis Category:Algebraic geometry Category:Differential topology