▁not ▁essentially ▁mis leading : ▁all ▁the ▁singular ities ▁of ▁algebraic ▁geometry ▁can ▁be ▁recovered ▁as ▁some ▁sort ▁of ▁very ▁general ▁collapse ▁( through ▁multiple ▁processes ). ▁This ▁result ▁is ▁often ▁implicitly ▁used ▁to ▁extend ▁aff ine ▁geometry ▁to ▁project ive ▁geometry : ▁it ▁is ▁entirely ▁typical ▁for ▁an ▁aff ine ▁variety ▁to ▁ac quire ▁singular ▁points ▁on ▁the ▁hyper plane ▁at ▁infinity , ▁when ▁its ▁closure ▁in ▁project ive ▁space ▁is ▁taken . ▁Resol ution ▁says ▁that ▁such ▁singular ities ▁can ▁be ▁handled ▁rather ▁as ▁a ▁( comp licated ) ▁sort ▁of ▁compact ification , ▁ending ▁up ▁with ▁a ▁compact ▁manifold ▁( for ▁the ▁strong ▁topology , ▁rather ▁than ▁the ▁Z aris ki ▁topology , ▁that ▁is ). ▁ ▁The ▁smooth ▁theory ▁and ▁cat ast roph es ▁At ▁about ▁the ▁same ▁time ▁as ▁Hir on aka ' s ▁work , ▁the ▁cat ast rop he ▁theory ▁of ▁René ▁Thom ▁was ▁receiving ▁a ▁great ▁deal ▁of ▁attention . ▁This ▁is ▁another ▁branch ▁of ▁singular ity ▁theory , ▁based ▁on ▁earlier ▁work ▁of ▁Hass ler ▁Whit ney ▁on ▁critical ▁points . ▁R ough ly ▁speaking , ▁a ▁critical ▁point ▁of ▁a ▁smooth ▁function ▁is ▁where ▁the ▁level ▁set ▁develop s ▁a ▁singular ▁point ▁in ▁the ▁geometric ▁sense . ▁This ▁theory ▁de als ▁with ▁differenti able ▁functions ▁in ▁general , ▁rather ▁than ▁just ▁polynomials . ▁To ▁compens ate , ▁only ▁the ▁stable ▁phen omena ▁are ▁considered . ▁One ▁can ▁argue ▁that ▁in ▁nature , ▁anything ▁destroyed ▁by ▁tiny ▁changes ▁is ▁not ▁going ▁to ▁be ▁observed