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о Бореловом опису, целобројна и мод 2 кохомологија многостру- кости застава дата је као полиномијална алгебра посечена по одређе- ном идеалу. У овом раду, Гребнерове базе за ове идеале добијене су у случају комплексних и реалних Грасманових многострукости, као и у случају реалних многострукости застава F(1,...,1; 2,...,2,k,n)... By Borel's description, integral and mod 2 cohomology of ag manifolds is a polynomial algebra modulo a well-known ideal. In this doctoral dissertation, Gr obner bases for these ideals are obtained in the case of complex and real Grassmann manifolds, and real ag manifolds F(1; : : : ; 1; 2; : : : ; 2; k; n). In the case of Grassmann manifolds, Gr obner bases are applied in the study of Z- cohomology of complex Grassmann manifolds. It is well-known that, besides Borel's description, this cohomology can be characterized in terms of Schubert classes. By establishing a connection between this description and Gr obner bases that we obtained, a new recurrence formula that can be used for calculating (all) Kostka numbers is derived. Using the same method for the small quantum cohomology of Grassmann manifolds (instead of the classical), these formulas are improved. In the case of real ag manifoldsF(1,...,1; 2,...,2,k,n), Gr obner bases are used to obtain new results on the immersions and embeddings of these manifolds, and for the calculation of the cup-length of some manifolds of this type. more...