No description available.

No description available.

Birational morphisms f: X $ to$ Y of nonsingular surfaces are studied first. Properties of the surfaces X and Y are shown to be related to certain numerical data extracted from the configuration of "missing curves" of f, that is, the curves in Y whose generic point is not in f (X). These results are then applied to the problem of decomposing birational endomorphisms of the plane into a succession of irreducible ones. / A graph-theoretic machinery is developed to keep track of the desingularization of the divisors at infinity of the plane. That machinery is then used to investigate the problem of classifying all birational endomorphisms of the plane, and a complete classification is given in the case of two fundamental points.

In this work we study the space of group homomorphisms Hom(Γ,G) from a geometric and simplicial point of view. The case in which the source group is a free abelian group of rank n is studied in more detail since this space can be identified with the space of commuting n-tuples of elements from G. This latter case is of particular interest when the target is a Lie group. The simplicial approach allows us to to construct a family of spaces that filters the classifying space of a group by filtering group theoretical information of the given group. Namely, we use the lower central series of free groups to construct a family of simplicial subspaces of the bar construction of the classifying space of a group. The first layer of this filtration is studied in more detail for transitively commutative (TC) groups.

No description available.

We construct classical tilting objects in derived categories of equivariant sheaves on quasi-projective varieties, which give equivalences with derived categories of modules over algebras. Our applications include a conceptual explanation of the importance of the McKay quiver associated to a representation of a finite group G and the development of a McKay correspondence for the cotangent bundle of the projective line. / Thesis (Ph.D, Mathematics & Statistics) -- Queen's University, 2008-09-04 14:42:25.099

No description available.

No description available.

This thesis consists of two parts, both being concerned with operads related to the ribbon braid groups. In the first part, we define a notion of semidirect product for operads and use it to study the framed $n$-discs operad (the semidirect product $f\mathcal{D}_n=\mathcal{D}_n\rtimes SO(n)$ of the little $n$-discs operad with the special orthogonal group). This enables us to deduce properties of $f\mathcal{D}_n$ from the corresponding properties for $\mathcal{D}_n$. We prove an equivariant recognition principle saying that algebras over the framed $n$-discs operad are $n$-fold loop spaces on $SO(n)$-spaces. We also study the operations induced on homology, showing that an $H(f\mathcal{D}_n)$-algebra is a higher dimensional Batalin-Vilkovisky algebra with some additional operators when $n$ is even. Contrastingly, for $n$ odd, we show that the Gerstenhaber structure coming from the little $n$-discs does not give rise to a Batalin-Vilkovisky structure. We give a general construction of operads from families of groups. We then show that the operad obtained from the ribbon braid groups is equivalent to the framed 2-discs operad. It follows that the classifying spaces of ribbon braided monoidal categories are double loop spaces on $S^1$-spaces. The second part of this thesis is concerned with infinite loop space structures on the stable mapping class group. Two such structures were discovered by Tillmann. We show that they are equivalent, constructing a map between the spectra of deloops. We first construct an `almost map', i.e a map between simplicial spaces for which one of the simplicial identities is satisfied only up to homotopy. We show that there are higher homotopies and deduce the existence of a rectification. We then show that the rectification gives an equivalence of spectra.

In this thesis we apply methods from homological algebra to the study of finite $p$-groups. Let $G$ be a finite $p$-group and let $\mathbb{F}_p$ be the field of $p$ elements. We consider the cohomology groups $\operatorname{H}^1(G,\mathbb{F}_p)$ and $\operatorname{H}^2(G,\mathbb{F}_p)$ and the Massey product structure on these cohomology groups, which we use to deduce properties about $G$. We tie the classical theory of Massey products in with a general method from deformation theory for constructing hulls of functors and see how far the strictly defined Massey products can take us in this setting. We show how these Massey products relate to extensions of modules and to relations, giving us cohomological presentations of $p$-groups. These presentations will be minimal pro-$p$ presentations and will often be different from the presentations we are used to. This enables us to shed some new light on the classification of $p$-groups, in particular we give a `tree construction' illustrating how we can `produce' $p$-groups using cohomological methods. We investigate groups of exponent $p$ and some of the families of groups appearing in the tree. We also investigate the limits of these methods. As an explicit example illustrating the theory we have introduced, we calculate Massey products using the Yoneda cocomplex and give 0-deficiency presentations for split metacyclic $p$-groups using strictly defined Massey products. We also apply these methods to the modular isomorphism problem, i.e. the problem whether (the isomorphism class of) $G$ is determined by $\F_pG$. We give a new class $\mathcal{C}$ of finite $p$-groups which can be distinguished using $\mathbb{F}_pG$.