When $G$ is abelian and $l$ is a prime we show how elements of the relative K-group $K_{0}({\bf Z}_{l}[G], {\bf Q}_{l})$ give rise to annihilator/Fitting ideal relations of certain associated ${\bf Z}[G]$-modules. Examples of this phenomenon are ubiquitous. Particularly, we give examples in which $G$ is the Galois group of an extension of global fields and the resulting annihilator/Fitting ideal relation is closely connected to Stickelberger's Theorem and to the conjectures of Coates and Sinnott, and Brumer. Higher Stickelberger ideals are defined in terms of special values of L-functions; when these vanish we show how to define fractional ideals, generalising the Stickelberger ideals, with similar annihilator properties. The fractional ideal is constructed from the Borel regulator and the leading term in the Taylor series for the L-function. En route, our methods yield new proofs, in the case of abelian number fields, of formulae predicted by Lichtenbaum for the orders of K-groups and étale cohomology groups of rings of algebraic integers.

Classification of real K3 surfaces $X$ with non-symplectic involution $\tau$ is considered. We define a natural notion of degeneration for them. We show that the connected component of moduli of non-degenerate surfaces of this type is defined by the isomorphism class of the action of $\tau$ and the anti-holomorphic involution $\varphi$ in the homology lattice. (There are very few similar results known.) For their classification we apply invariants of integral lattice involutions with conditions that were developed by the first author in 1983. As a particular case, we describe connected components of moduli of real non-singular curves $A \in | -2 K_V|$ for the classical real surfaces: $V = P^2$, hyperboloid, ellipsoid, $F_1$, $F_4$.

As an application, we describe all real polarized K3 surfaces that are deformations of general real K3 double rational scrolls (the surfaces $V$ above). There are very few exceptions. For example, any non-singular real quartic in $P^3$ can be constructed in this way.

We study the relationship between the cohomology of $q$-Schur algebras and Hecke algebras in type $A$. A key tool, which has independent interest, involves a specific Hecke algebra resolution, introduced in a special case by Deodhar. This resolution is somewhat akin to the Koszul resolution for Lie algebra cohomology, in that it has an explicit description yet applies for all modules. An application of the properties of this resolution establishes a ‘quantum Weyl reciprocity’ connecting, in a range of degrees, the cohomology of the $q$-Schur algebra with that of the Hecke algebra. This work also provides, independently of the resolution, cohomological inequalities which apply in all degrees and which sometimes can be related to Kazhdan–Lusztig polynomials. Finally, we extend to cohomology a decomposition number method of Erdmann, and illustrate its usefulness in computations. This method enables us to exhibit some properties of symmetric group cohomology which, if true, would imply the $SL_n$-version of Lusztig's characteristic $p$ character conjecture.

Let $G$ be a semisimple Lie group of $\mathbb{R}$-rank at least 2 and $\varGamma$ a discrete subgroup of $G$. We consider the limit set of $\varGamma$ in the geometric boundary of the symmetric space associated with $G$. We define the notion of conical and horospherical limit points. In the case of irreducible non-uniform lattices, by using the two Tits building structures, we distinguish the location of their conical limit points. The limit sets of generalized Schottky groups contained in Hilbert modular groups are studied.

We introduce a notion of a category with full projective functors. It encodes certain common properties of categories appearing in representation theory of Lie groups, Lie algebras and quantum groups. We describe the left or right exact functors which naturally commute with projective functors and provide a unified approach to the verification of relations between such functors.

Two locally generic maps $f, g \colon M^n \to \mathbb{R}^{2n - 1}$ are regularly homotopic if they lie in the same path-component of the space of locally generic maps. Our main result is that if $n \neq 3$ and $M^n$ is a closed $n$-manifold then the regular homotopy class of every locally generic map $f \colon M^n \to \mathbb{R}^{2n - 1}$ is completely determined by the number of its singular points provided that $f$ is singular (that is, $f$ is not an immersion).

We construct an orientable ribbon surface $F \subset B^4$ which is universal in the following sense: any orientable 4-manifold $M \cong B^4 \cup \text{1-handles} \cup \text{2-handles}$ can be represented as a cover of $B^4$ branched over $F$.

In this paper, we study applications of Atiyah–Hirzebruch spectral sequences for motivic cobordism recently found by Hopkins and Morel. For example, we compute the mod 2 motivic cohomology of a reduced Cech complex of a splitting variety according to Orlov, Vishik and Voevodsky. The cobordism ring $MGL / 2^{*,*} (Spec(\mathbb{R}))$ is computed and compared with the results of the real cobordism theory of Hu and Kriz. Moreover, we study algebraic cobordism of the classifying spaces $BG$ for algebraic groups over $\mathbb{C}$. For example, the Chow ring $CH^* (BG_2)_(2)$ for the exceptional Lie group $G_2$ is determined by using motivic cobordism and motivic cohomology.