▁of ▁convex ▁real ▁project ive ▁structures ▁on ▁a ▁closed ▁orient able ▁surface ▁of ▁genus ▁ ▁is ▁home omorphic ▁to ▁an ▁open ▁cell ▁of ▁dimension ▁. ▁With ▁Su hy ou ng ▁Ch oi , ▁he ▁proved ▁that ▁this ▁space ▁is ▁a ▁connected ▁component ▁( the ▁" H itch in ▁component ") ▁of ▁the ▁space ▁of ▁equivalence ▁classes ▁of ▁representations ▁of ▁the ▁fundamental ▁group ▁in ▁. ▁ ▁Com bin ing ▁this ▁with ▁Su hy ou ng ▁Ch oi ' s ▁convex ▁decomposition ▁theorem , ▁this ▁led ▁to ▁a ▁complete ▁classification ▁of ▁convex ▁real ▁project ive ▁structures ▁on ▁compact ▁surfaces . ▁ ▁His ▁doctor al ▁dis sert ation , ▁" Dis continu ous ▁groups ▁and ▁the ▁E uler ▁class " ▁( super vised ▁by ▁Morris ▁W . ▁Hir sch ), ▁character izes ▁discrete ▁embed d ings ▁of ▁surface ▁groups ▁in ▁ ▁in ▁terms ▁of ▁maximal ▁E uler ▁class , ▁prov ing ▁a ▁con verse ▁to ▁the ▁Mil nor – W ood ▁inequality ▁for ▁flat ▁bund les . ▁Short ly ▁there after ▁he ▁showed ▁that ▁the ▁space ▁of ▁representations ▁of ▁the ▁fundamental ▁group ▁of ▁a ▁closed ▁orient able ▁surface ▁of ▁genus ▁ ▁in ▁ ▁has ▁ ▁connected ▁components , ▁distinguished ▁by ▁the ▁E uler ▁class . ▁ ▁With ▁David ▁Fried , ▁he ▁class ified ▁compact ▁quot ients ▁of ▁E uclidean ▁ 3 - space ▁by ▁discrete ▁groups ▁of ▁aff ine ▁transformations , ▁showing ▁that ▁all ▁such ▁manif olds ▁are ▁finite ▁quot ients ▁of ▁tor us ▁bund les ▁over ▁the ▁circle . ▁The ▁non comp act ▁case ▁is