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For $n\in\{2^t-3,2^t-2,2^t-1\}$ ($t\ge3$) we study the cohomology algebra $H^*(\widetilde G_{n,3};\mathbb Z_2)$ of the Grassmann manifold $\widetilde G_{n,3}$ of oriented $3$-dimensional subspaces of $\mathbb R^n$. A complete description of $H^*(\widetilde G_{n,3};\mathbb Z_2)$ is given in the cases $n=2^t-3$ and $n=2^t-2$, while in the case $n=2^t-1$ we obtain a description complete up to a coefficient from $\mathbb Z_2$., Comment: 15 pages

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.

Predmet ove doktorske disertacije je primena algebarskih tehnika na jednu od centralnih tema kombinatorike i diskretne geometrije - poliomino poplocavanja. Poliomino poplocavanja su interesantna ne samo matematicarima nego i zicarima i biolozima, a imaju i primenu u racunarskim naukama. U ovoj disertaciji akcenat je stavljen na mogucnost da se posebna klasa problema poplocavanja koja su invarijantna u odnosu na delovanje konacne grupe resi primenom teorije Grebnerovih baza za prstene polinoma nad prstenom celih brojeva Z. Metoda koja se koristi odrazava duboku povezanost izmedu algebre, geometrije i kombinatorike... Subject of this doctoral thesis is the application of algebraic techniques on one of the central topics of combinatorics and discrete geometry - polyomino tiling. Polyomino tilings are interesting not only to mathematicians, but also to physicists and biologists, and they can also be applied in computer science. In this thesis we put some emphasis on possibility to solve special class of tiling problems, that are invariant under the action of nite group, by using theory of Gr obner basis for polynomial rings with integer coecients. Method used here re ects deep connection between algebra, geometry and combinatorics...

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.