Editing Singularity theory (section)

==The smooth theory and catastrophes==
At about the same time as Hironaka's work, the [[catastrophe theory]] of [[René Thom]] was receiving a great deal of attention. This is another branch of singularity theory, based on earlier work of [[Hassler Whitney]] on [[critical point (mathematics)|critical point]]s. Roughly speaking, a ''critical point'' of a [[smooth function]] is where the [[level set]] develops a singular point in the geometric sense. This theory deals with differentiable functions in general, rather than just polynomials. To compensate, only the ''stable'' phenomena are considered. One can argue that in nature, anything destroyed by tiny changes is not going to be observed; the visible ''is'' the stable. Whitney had shown that in low numbers of variables the stable structure of critical points is very restricted, in local terms. Thom built on this, and his own earlier work, to create a ''catastrophe theory'' supposed to account for discontinuous change in nature.

===Arnold's view===
While Thom was an eminent mathematician, the subsequent fashionable nature of elementary [[catastrophe theory]] as propagated by [[Christopher Zeeman]] caused a reaction, in particular on the part of [[Vladimir Arnold]].<ref>{{harvnb|Arnold|1992}}</ref> 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 relation]]s on singular points, and [[germ (mathematics)|germs]]. Technically this involves [[Group action (mathematics)|group action]]s of [[Lie group]]s on spaces of [[jet (mathematics)|jet]]s; in less abstract terms [[Taylor series]] are examined up to change of variable, pinning down singularities with enough [[derivative]]s. Applications, according to Arnold, are to be seen in [[symplectic geometry]], as the geometric form of [[classical mechanics]].

===Duality===
An important reason why singularities cause problems in mathematics is that, with a failure of manifold structure, the invocation of [[Poincaré duality]] is also disallowed. A major advance was the introduction of [[intersection cohomology]], which arose initially from attempts to restore duality by use of strata. Numerous connections and applications stemmed from the original idea, for example the concept of [[perverse sheaf]] in [[homological algebra]].