▁part ▁of ▁modern ▁mathematics ▁and ▁which ▁he ▁was ▁for em ost ▁in ▁sh aping ▁and ▁adv ancing . ▁This ▁field ▁cent ers ▁on ▁Lie ▁groups , ▁partial ▁differential ▁systems , ▁and ▁differential ▁geometry ; ▁these , ▁chief ly ▁through ▁Cart an ' s ▁contributions , ▁are ▁now ▁closely ▁inter w oven ▁and ▁const itute ▁a ▁un ified ▁and ▁powerful ▁tool . ▁ ▁Lie ▁groups ▁Cart an ▁was ▁pract ically ▁alone ▁in ▁the ▁field ▁of ▁Lie ▁groups ▁for ▁the ▁thirty ▁years ▁after ▁his ▁dis sert ation . ▁Lie ▁had ▁considered ▁these ▁groups ▁chief ly ▁as ▁systems ▁of ▁analyt ic ▁transformations ▁of ▁an ▁analyt ic ▁manifold , ▁depending ▁analyt ically ▁on ▁a ▁finite ▁number ▁of ▁parameters . ▁A ▁very ▁fruit ful ▁approach ▁to ▁the ▁study ▁of ▁these ▁groups ▁was ▁opened ▁in ▁ 1 8 8 8 ▁when ▁Wilhelm ▁K illing ▁system atically ▁started ▁to ▁study ▁the ▁group ▁in ▁itself , ▁independent ▁of ▁its ▁possible ▁actions ▁on ▁other ▁manif olds . ▁At ▁that ▁time ▁( and ▁until ▁ 1 9 2 0 ) ▁only ▁local ▁properties ▁were ▁considered , ▁so ▁the ▁main ▁object ▁of ▁study ▁for ▁K illing ▁was ▁the ▁Lie ▁algebra ▁of ▁the ▁group , ▁which ▁exactly ▁reflect s ▁the ▁local ▁properties ▁in ▁purely ▁algebraic ▁terms . ▁K illing ' s ▁great ▁achiev ement ▁was ▁the ▁determ ination ▁of ▁all ▁simple ▁complex ▁Lie ▁al gebras ; ▁his ▁proofs , ▁however , ▁were ▁often ▁defect ive , ▁and ▁Cart an ' s ▁th esis ▁was ▁devoted ▁mainly ▁to ▁giving ▁a ▁rig orous ▁foundation ▁to ▁the ▁local