René Thom: Difference between revisions

While René Thom is most known to the public for his development of [[catastrophe theory]] between 1968 and 1972, his earlier work was on [[differential topology]]. In the early 1950s it concerned what are now called [[Thom space]]s, [[characteristic class]]es, [[cobordism theory]], and the [[Thom transversality theorem]]. Another example of this line of work is the [[Thom conjecture]], versions of which have been investigated using [[gauge theory]]. From the mid 1950s he moved into [[singularity theory]], of which catastrophe theory is just one aspect, and in a series of deep (and at the time obscure) papers between 1960 and 1969 developed the theory of [[topologically stratified space|stratified sets]] and stratified maps, proving a basic stratified isotopy theorem describing the local conical structure of [[Whitney conditions|Whitney stratified sets]], now known as the [[Thom–Mather isotopy theorem]]. Much of his work on stratified sets was developed so as to understand the notion of topologically stable maps, and to eventually prove the result that the set of topologically stable mappings between two smooth manifolds is a [[dense set]]. Thom's lectures on the stability of differentiable mappings, given at the [[University of Bonn]] in 1960, were written up by [[Harold Levine]] and published in the proceedings of a year long symposium on singularities at [[Liverpool University]] during 1969-70, edited by [[C. T. C. Wall]]. The proof of the density of topologically stable mappings was completed by [[John Mather (mathematician)|John Mather]] in 1970, based on the ideas developed by Thom in the previous ten years. A coherent detailed account was published in 1976 by Christopher Gibson, Klaus Wirthmüller, Andrew du Plessis, and [[Eduard Looijenga]].