Abstract not available. / text

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.

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.

No description available.

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.

: La théorie géométrique des invariants constitue un domaine central de la géométrie algébrique d'aujourd'hui : développée par Mumford au début des années soixante, elle a conduit à des progrès considérables dans l'étude des variétés projectives, notamment par la construction d'espaces de modules. Dans les vingt dernières années des interactions entre la théorie géométrique des invariants et la géométrie arithmétique -- plus précisément la théorie des hauteurs et la géométrie d'Arakelov -- ont été étudiés par divers auteurs (Burnol, Bost, Zhang, Soulé, Gasbarri, Chen). Dans cette thèse nous nous proposons d'un côté d'étudier de manière systématique la théorie géométrique des invariants dans le cadre de la géométrique d'Arakelov ; de l'autre de montrer que ces résultats permettent une nouvelle approche géométrique (distincte aussi de la méthode des pentes développée par Bost) aux résultats d'approximation diophantienne, tels que le Théorème de Roth et ses généralisations par Lang, Wirsing et Vojta. / Geometric invariant theory is a central subject in nowadays' algebraic geometry : developed by Mumford in the early sixties, it enhanced the knowledge of projective varieties through the construction of moduli spaces. During the last twenty years, interactions between geometric invariant theory and arithmetic geometric --- more precisely, height theory and Arakelov geometry --- have been exploited by several authors (Burnol, Bost, Zhang, Soulé, Gasbarri, Chen). In this thesis we firstly study in a systematic way how geometric invariant theory fits in the framework of Arakelov geometry; then we show that these results give a new geometric approach to questions in diophantine approximation, proving Roth's Theorem and its recent generalizations by Lang, Wirsing and Vojta.

Human motion analysis is one of the major problems in computer vision research. It deals with the study of the motion of human body in video data from different aspects, ranging from the tracking of body parts and reconstruction of 3D human body configuration, to higher level of interpretation of human action and activities in image sequences. When human motion is observed through video camera, it is perspectively distorted and may appear totally different from different viewpoints. Therefore it is highly challenging to establish correct relationships between human motions across video sequences with different camera settings. In this work, we investigate the geometric invariance in the motion of human body, which is critical to accurately understand human motion in video data regardless of variations in camera parameters and viewpoints. In human action analysis, the representation of human action is a very important issue, and it usually determines the nature of the solutions, including their limits in resolving the problem. Unlike existing research that study human motion as a whole 2D/3D object or a sequence of postures, we study human motion as a sequence of body pose transitions. We also decompose a human body pose further into a number of body point triplets, and break down a pose transition into the transition of a set of body point triplets. In this way the study of complex non-rigid motion of human body is reduced to that of the motion of rigid body point triplets, i.e. a collection of planes in motion. As a result, projective geometry and linear algebra can be applied to explore the geometric invariance in human motion. Based on this formulation, we have discovered the fundamental ratio invariant and the eigenvalue equality invariant in human motion. We also propose solutions based on these geometric invariants to the problems of view-invariant recognition of human postures and actions, as well as analysis of human motion styles. These invariants and their applicability have been validated by experimental results supporting that their effectiveness in understanding human motion with various camera parameters and viewpoints.

Les coefficients de Kronecker, qui sont indexés par des triplets de partitions et décrivent la décomposition du produit tensoriel de deux représentations irréductibles d'un groupe symétrique en somme directe de telles représentations, ont été introduits par Francis Murnaghan dans les années 1930. Il a notamment remarqué un comportement particulier de ces coefficients : à partir de n'importe quel triplet de partitions, on peut construire une certaine suite de coefficients de Kronecker qui est stationnaire.Afin de généraliser cette propriété, John Stembridge a introduit en 2014 une notion de stabilité pour les triplets de partitions, ainsi qu'une autre notion -- celle de triplet faiblement stable -- dont il a conjecturé qu'elle serait équivalente à la précédente. Cette conjecture a été démontrée peu après par Steven Sam et Andrew Snowden, par des méthodes algébriques.Dans cette thèse, on donne notamment une autre démonstration -- cette fois géométrique -- de cette équivalence grâce à l'interprétation classique des coefficients de Kronecker comme dimensions d'espaces de sections de fibrés en droites sur des variétés de drapeaux. Ces méthodes permettent également de s'intéresser à quelques questions plus précises : la stabilité dont on parle consiste en le fait que certaines suites de coefficients sont stationnaires, et on se demande à partir de quand ces suites deviennent constantes.On applique ensuite ces techniques à d'autres exemples de coefficients de branchement, puis on s'intéresse à un autre problème : celui de produire des triplets stables de partitions. On généralise ainsi un résultat obtenu indépendamment par Laurent Manivel et Ernesto Vallejo sur ce sujet / The Kronecker coefficients, which are indexed by triples of partitions and describe how the tensor product of two irreducible representations of the symmetric group decomposes as a direct sum of such representations, were introduced by Francis Murnaghan in the 1930s. He notably noticed a remarkable behaviour of these coefficients: from any triple of partitions, one can construct a particular sequence of Kronecker coefficients which eventually stabilises.In order to generalise this property, John Stembridge introduced in 2014 a notion of stability for triples of partitions, as well as another notion -- of weakly stable triple -- about which he conjectured that it should be equivalent to the previous one. This conjecture was proven shortly after by Steven Sam and Andrew Snowden, with algebraic methods.In this thesis we especially give another proof -- this time geometric -- of this equivalence, using the classical expression of the Kronecker coefficients as dimensions of spaces of sections of line bundles on flag varieties. With these methods we can also be interested in more specific questions: since the stability which we discuss means that some sequences of coefficients stabilise, one can wonder at which point these sequences become constant.We then apply these techniques to other examples of branching coefficients, and are also interested in another problem: how can we produce stable triples of partitions? We thus generalise a result obtained independently by Laurent Manivel and Ernesto Vallejo on this subject