Singularity theory

For non-mathematical singularity theories, see singularity.

Arnol'd's view

While Thom was an eminent mathematician, the subsequent fashionable nature of elementary catastrophe theory caused a reaction, in particular on the part of Vladimir Arnol'd. He may have been largely responsible for applying the term singularity theory to the area including the input from algebraic geometry, as well as that flowing from the work of Whitney, Thom and other authors. He wrote in terms making clear his distaste for the too-publicised emphasis on a small part of the territory. The foundational work on smooth singularities is formulated as the construction of equivalence relations on singular points, and germs. Technically this involves group actions of Lie groups on spaces of jets; in less abstract terms Taylor series are examined up to change of variable, pinning down singularities with enough derivatives. Applications, according to Arnol'd, are to be seen in symplectic geometry, as the geometric form of classical mechanics.

Related Topics:
Elementary catastrophe theory - Vladimir Arnol'd - Equivalence relation - Germs - Group action - Lie group - Jet - Taylor series - Derivative - Symplectic geometry - Classical mechanics

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