The (small) quantum cohomology ring of a Grassmann variety encodes the enumerative geometry of rational curves in this variety. By using degeneracy loci formulas on quot schemes, Bertram has proved quantum Pieri and Giambelli formulas which give a complete description of the quantum cohomology ring. In this talk I will present elementary new proofs of these results which rely only on the defini...

The ring structure of the integral cohomology of complements of real linear subspace arrangements is considered. While the additive structure of the cohomology is given in terms of the intersection poset and dimension function by a theorem of Goresky and MacPherson, we describe the multiplicative structure in terms of the intersection poset, the dimension function and orientations of the partic...

where In is the ideal generated by symmetric polynomials in x1, . . . , xn without constant term. Another, geometric, description of the cohomology ring of the flag manifold is based on the decomposition of Fln into Schubert cells. These are even-dimensional cells indexed by the elements w of the symmetric group Sn . The corresponding cohomology classes σw , called Schubert classes, form an add...

where In is the ideal generated by symmetric polynomials in x1, . . . , xn without constant term. Another, geometric, description of the cohomology ring of the flag manifold is based on the decomposition of Fln into Schubert cells. These are even-dimensional cells indexed by the elements w of the symmetric group Sn . The corresponding cohomology classes σw , called Schubert classes, form an add...

Let $R = bigoplus_{n in mathbb{N}_{0}} R_{n}$ be a standardgraded ring, $M$ be a finitely generated graded $R$-module and $J$be a homogenous ideal of $R$. In this paper we study the gradedstructure of the $i$-th local cohomology module of $M$ defined by apair of ideals $(R_{+},J)$, i.e. $H^{i}_{R_{+},J}(M)$. Moreprecisely, we discuss finiteness property and vanishing of thegraded components $H^...

The Bianchi groups are a family of discrete subgroups of PSL2(C) which have group theoretic descriptions as amalgamated products and HNN extensions. Using Bass-Serre theory, we show how the cohomology of these two constructions relates to the cohomology of their pieces. We then apply these results to calculate the mod-2 cohomology ring for various Bianchi groups.

We construct the q-analogue of a certain class of group cohomology and introduce the action of Hecke operators on such cohomology. We also show that such an action determines a representation of a Hecke ring in each of the associated group cohomology spaces. MSC 2000: 20G10, 11F60 (primary); 11F75, 18G99 (secondary)

We define abelian extensions of algebras in congruence-modular varieties. The theory is sufficiently general that it includes, in a natural way, extensions of R-modules for a ring R. We also define a cohomology theory, which we call clone cohomology, such that the cohomology group in dimension one is the group of equivalence classes of extensions.

We study the topology of the boundary manifold of a regular neighborhood of a complex projective hypersurface. We show that, under certain Hodge theoretic conditions, the cohomology ring of the complement of the hypersurface functorially determines that of the boundary. When the hypersurface defines a hyperplane arrangement, the cohomology of the boundary is completely determined by the combina...

In this note we describe aspects of the cohomology of coherent sheaves on a complete toric variety X over a field k and, more generally, the local cohomology, with supports in a monomial ideal, of a finitely generated module over a polynomial ring S. This leads to an efficient way of computing such cohomology, for which we give explicit algorithms. The problem is finiteness. The i local cohomol...