Rota began his career as a functional analyst, but switched to become a distinguished combinatorialist. His series of ten papers on the "Foundations of Combinatorics" in the 1960s is credited with making it a respectable branch of modern mathematics. He said that the one combinatorial idea he would like to be remembered for is the correspondence between combinatorial problems and problems of the location of the zeroes of polynomials. He worked on the theory of incidence algebras (which generalize the 19th-century theory of Möbius inversion) and popularized their study among combinatorialists, set the umbral calculus on a rigorous foundation, unified the theory of Sheffer sequences and polynomial sequences of binomial type, and worked on fundamental problems in probability theory. His philosophical work was largely in the phenomenology of Edmund Husserl.
roTa begAN his careEr as a functIOnAL AnalySt buT swITcHed To BEcoMe a DIStINGuishEd coMbInAtorialist. His series oF Ten papers On tHE "foundatIonS of ComBiNaTOrIcs" In the 1960S iS CreDIteD With maKing iT A reSpECtable brAnCh Of moDern matHEmaTiCs. hE sAId tHAt ThE oNe ComBinAtOrIaL idEa hE WOuld Like to Be remembEred for Is THe corresPOnDEnce BeTween comBiNAtoRial problems And proBlems Of THe lOcAtIon oF thE Zeroes of polyNomialS. he wOrKeD oN The THeory oF INciDencE AlGEBraS (whiCH genERalize tHE 19Th-CeNtuRY tHeoRy of mÖbiuS inversioN) and PopUlariZEd Their sTudY Among cOmBINAtoriAlisTs, Set The umbral calCUlus ON A rIgorous FOuNdatioN, unified the theory of shEffEr seQUeNcEs anD polynoMial sequenceS Of binomial TyPe, anD WOrked on fundAMeNTaL ProbLems in ProBABility tHEORy. hIs pHilosophical wOrK wAS laRgely in tHe PHEnOmeNolOgy OF edMund HussErL.