▁geometry , ▁as ▁the ▁geometric ▁form ▁of ▁classical ▁mechan ics . ▁ ▁D ual ity ▁An ▁important ▁reason ▁why ▁singular ities ▁cause ▁problems ▁in ▁mathematics ▁is ▁that , ▁with ▁a ▁failure ▁of ▁manifold ▁structure , ▁the ▁inv ocation ▁of ▁Po inc ar é ▁du ality ▁is ▁also ▁dis allowed . ▁A ▁major ▁advance ▁was ▁the ▁introduction ▁of ▁intersection ▁coh om ology , ▁which ▁arose ▁initially ▁from ▁attempts ▁to ▁restore ▁du ality ▁by ▁use ▁of ▁str ata . ▁N umer ous ▁connections ▁and ▁applications ▁stem med ▁from ▁the ▁original ▁idea , ▁for ▁example ▁the ▁concept ▁of ▁per verse ▁she af ▁in ▁hom ological ▁algebra . ▁ ▁Other ▁possible ▁mean ings ▁The ▁theory ▁mentioned ▁above ▁does ▁not ▁directly ▁relate ▁to ▁the ▁concept ▁of ▁mathematical ▁singular ity ▁as ▁a ▁value ▁at ▁which ▁a ▁function ▁is ▁not ▁defined . ▁For ▁that , ▁see ▁for ▁example ▁isolated ▁singular ity , ▁essential ▁singular ity , ▁remov able ▁singular ity . ▁The ▁mon od rom y ▁theory ▁of ▁differential ▁equations , ▁in ▁the ▁complex ▁domain , ▁around ▁singular ities , ▁does ▁however ▁come ▁into ▁relation ▁with ▁the ▁geometric ▁theory . ▁R ough ly ▁speaking , ▁mon od rom y ▁studies ▁the ▁way ▁a ▁covering ▁map ▁can ▁deg ener ate , ▁while ▁singular ity ▁theory ▁studies ▁the ▁way ▁a ▁manifold ▁can ▁deg ener ate ; ▁and ▁these ▁fields ▁are ▁linked . ▁ ▁See ▁also ▁ ▁Tang ent ▁Z aris ki ▁tang ent ▁space ▁General ▁position ▁Contact ▁( math emat ics ) ▁Sing ular ▁solution ▁Str at ification ▁( math emat ics