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Abstract:
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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 manifolds F(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. |