This thesis is on moduli spaces of complexes of sheaves and diagrams of such moduli spaces. The objects in these diagrams are constructed as geometric invariant theory quotients and the points in these quotients correspond to certain equivalence classes of complexes. The morphisms in these diagrams are constructed by taking direct sums with acyclic complexes. We then study the colimit of such a diagram and in particular are interested in studying the images of quasi-isomorphic complexes in the colimit. As part of this thesis we construct categorical quotients of a group action on unstable strata appearing in a stratification associated to a complex projective scheme with a reductive group action linearised by an ample line bundle. We study this stratification for a closed subscheme of a quot scheme parametrising quotient sheaves over a complex projective scheme and relate the Harder-Narasimhan types of unstable sheaves with the unstable strata in the associated stratification. We also study the stratification of a parameter space for complexes with respect to a linearisation determined by certain stability parameters and show that a similar result holds in this case. The objects in these diagrams are indexed by different Harder-Narasimhan types for complexes and are quotients of parameter schemes for complexes of this fixed Harder-Narasimhan type. This quotient is given by a choice of linearisation of the action and so the diagrams depend on these choices. We conjecture that these choices can be made so that for any quasi-isomorphism between complexes representing points in this diagram both complexes are identified in the colimit of this diagram.

The topic of this thesis is the moduli theory of (parabolic) sheaves on stable curves. Using geometric invariant theory (GIT), universal moduli spaces of semistable parabolic sheaves on stable marked curves are constructed: `universal' indicates that these are moduli spaces of pairs where the underlying marked curve may vary as well as the parabolic sheaf (as in the Pandharipande moduli space for pairs of stable curves and torsion-free sheaves without augmentations). As an intermediate step in this construction, we construct moduli spaces of semistable parabolic sheaves on flat families of arbitrary projective schemes (of any dimension or singularity type): this is the technical core of this thesis. These moduli spaces are projective, since they are constructed as GIT quotients of projective parameter spaces. The stability condition for parabolic sheaves depends on a choice of polarisation and is derived from the Hilbert-Mumford criterion. It is not quite the same as traditional stability with respect to parabolic Hilbert polynomials, but it is closely related to it, and the resulting moduli spaces are always compactifications of moduli of slope-stable parabolic sheaves. The construction works over algebraically closed fields of arbitrary characteristic.

Suppose that G is a finite, unitary reflection group acting on a complex vector space V and X is a subspace of V. Define N to be the setwise stabilizer of X in G, Z to be the pointwise stabilizer, and C=N/Z. Then restriction defines a homomorphism from the algebra of G-invariant polynomial functions on V to the algebra of C-invariant functions on X. In my thesis, I extend earlier work by Douglass and Röhrle for Coxeter groups to the case where G is a complex reflection group of type G(r,p,n) in the notation of Shephard and Todd and X is in the lattice of the reflection arrangement of G. The main result characterizes when the restriction mapping is surjective in terms of the exponents of G and C and their reflection arrangements.

Neste trabalho, introduzimos a teoria invariante e equivariante para a ação do grupo de Lorentz no espaço de Minkowski. Na teoria clássica, muitos resultados são válidos somente para a ação de grupos compactos em espaços Euclideanos. Continuamos o estudo para alguns subgrupos de Lorentz compactos e apresentamos uma forma de calcular as involuções de Lorentz em O(n;1). Fazemos uma empolgante discussão sobre uma classe de matrizes centrossimétricas polinomiais com aplicações em teoria invariante, estabelecendo um rumo para a pesquisa em subgrupos de Lorentz não compactos. Por fim, apresentamos alguns resultados da teoria equivariante para subgrupos de Lorentz. / In this work, we introduce the invariant and equivariant theory for the Lorentz group on the Minkowski space. In the classical theory, many results are valid only for compact groups on Euclidean spaces. We continue the study of some compact Lorentz subgroups and present a way of calculating the Lorentz involutions in O(n;1). We make an exciting discussion about a class of polynomial centrosymmetric matrices with applications in invariant theory, setting a course for research in non-compact Lorentz groups. Finally, we present some results for the equivariant theory of Lorentz subgroups.

On s'intéresse aux invariants pour l'action naturelle du groupe SL_2<br />sur l'algèbre B des coordonnées homogènes de la Grassmannienne des<br />pinceaux de formes quintiques binaires. La variété quotient<br />Proj(B^SL_2) est un candidat naturel pour la variété de modules des<br />quintiques gauches rationnelles.<br /><br />Un procédé connu établit une correspondance birationnelle et<br />équivariante entre la Grassmannienne des pinceaux de formes binaires<br />de degré d et l'espace projectif des formes binaires de degré 2d-2.<br />Lorsque le degré d est 5, cela suggère de comparer l'algèbre B^SL_2 et<br />l'algèbre des invariants d'une forme octique binaire. Cette algèbre a<br />été décrite en détail par T. Shioda en 1967.<br /><br />Nous établissons pour B^SL_2 un résultat analogue à celui de T.<br />Shioda : l'algèbre B^SL_2 est le quotient de l'algèbre de polynômes à<br />neuf indéterminées R=C[x_1,x_2,x_3,x'_3,x_4,x_5,x'_5,x_6,x_7] (les<br />indices donnent les degrés des indéterminées) par l'idéal des<br />4-Pfaffiens d'une matrice alternée 5x5 ; on identifie (numériquement)<br />la résolution libre minimale du R-module B^SL_2 ; enfin, on obtient<br />une famille génératrice minimale de l'algèbre B^SL_2.<br /><br />Pour y parvenir on commence par étendre la formule de T. Springer<br />(donnant la série de Poincaré de l'algèbre des invariants d'une forme<br />binaire) à l'algèbre des coordonnées homogènes d'une Grassmannienne.<br /><br /><br />Le point clé suivant consiste en l'identification d'un système de<br />paramètres homogènes. C'est possible grâce à une caractérisation, au<br />moyen du morphisme Wronskien, de la stabilité sur la Grassmannienne.<br />Il faut ensuite étudier les covariants d'ordre 4 et degré 2, ce qui<br />donne lieu à quelques énoncés de nature géométrique.<br /><br />Ces techniques permettent également de décrire les algèbres<br />d'invariants des pinceaux de cubiques et quartiques. Par ailleurs<br />l'étude du Wronskien conduit à de nouvelles formules de pléthysme.

Superintegrable systems are classical and quantum Hamiltonian systems which enjoy much symmetry and structure that permit their solubility via analytic and even, algebraic means. The problem of classification of superintegrable systems can be approached by considering associated geometric structures. To this end, we invoke the invariant theory of Killing tensors (ITKT), and the recursive version of the Cartan method of moving frames to derive joint invariants. We are able to intrinsically characterise and interpret the arbitrary parameters appearing in the general form of the Smorodinsky-Winternitz superintegrable potential, where we determine that the more general the geometric structure associated with the SW potential is, the fewer arbitrary parameters it admits. Additionally, we classify the multi-separability of the Tremblay-Turbiner-Winternitz (TTW) system. We provide a proof that only for the case k = +/- 1 does the general TTW system admit orthogonal separation of variables with respect to both Cartesian and polar coordinates. / A study towards the classification of superintegrable systems defined on the Euclidean plane.

No description available.

Neste trabalho, introduzimos a teoria invariante e equivariante para a ação do grupo de Lorentz no espaço de Minkowski. Na teoria clássica, muitos resultados são válidos somente para a ação de grupos compactos em espaços Euclideanos. Continuamos o estudo para alguns subgrupos de Lorentz compactos e apresentamos uma forma de calcular as involuções de Lorentz em O(n;1). Fazemos uma empolgante discussão sobre uma classe de matrizes centrossimétricas polinomiais com aplicações em teoria invariante, estabelecendo um rumo para a pesquisa em subgrupos de Lorentz não compactos. Por fim, apresentamos alguns resultados da teoria equivariante para subgrupos de Lorentz. / In this work, we introduce the invariant and equivariant theory for the Lorentz group on the Minkowski space. In the classical theory, many results are valid only for compact groups on Euclidean spaces. We continue the study of some compact Lorentz subgroups and present a way of calculating the Lorentz involutions in O(n;1). We make an exciting discussion about a class of polynomial centrosymmetric matrices with applications in invariant theory, setting a course for research in non-compact Lorentz groups. Finally, we present some results for the equivariant theory of Lorentz subgroups.

The realizations of the basic representation of the affine general linear Lie algebra on (r x r) matrices are well-known to be parametrized by partitions of r and have an explicit description in terms of vertex operators on the bosonic/fermionic Fock space. In this thesis, we give a geometric interpretation of these realizations in terms of geometric operators acting on the equivariant cohomology of certain Nakajima quiver varieties.

No description available.