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...Spring research interactions in mathematics note for periodic table topological insulators and superconductors en jui kuo department of physics university maryland college park u s e mail kuoenjui umd edu abstract this i try to show import ingredients how kitaev gave the provide relevant mathematical background including algebraic topol ogy higher homotopy bott periodicity clifford algebra symmetric space k theory mostly followed martin r zirnbauer lecture notes allen hatcher topology other figure contents introduction symmetricspaceandclassicationofhamiltonian symmetryspace bottperiodicity loop periodictable conclusion a basicintroduction b toolofcalculating group c classication algebras real complex d little explained why leads intense activity on started about years ago after theoret ical experimental discovery quantum spin hall insulator as name suggests is close cousin standard effect although it differs from by presence time reversal symmetry meaning invariance under hypothetical...