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Gabriel's Theorem describes the dimension vectors of the finitely generated indecomposable modules over the path algebra of a simply-laced Dynkin quiver. It shows that they can be obtained from the expressions for the positive roots of the corresponding root system in terms of the simple roots. Here, we present a method for finding the dimension vectors of the finitely generated indecomposable modules over a cluster-tilted algebra of Dynkin type A.;It is known that the quiver of a cluster-tilted algebra of Dynkin type A is given by an exchange matrix of the corresponding cluster algebra. We define a companion basis for such a quiver to be a Z -basis of roots of the integral root lattice of the corresponding root system whose associated matrix of inner products is a positive quasi-Cartan companion of the corresponding exchange matrix.;Our main result establishes that the dimension vectors of the finitely generated indecomposable modules over a cluster-tilted algebra of Dynkin type A arise from expressions for the positive roots of the corresponding root system in terms of a companion basis (for the quiver of that algebra). This can be regarded as a generalisation of part of Gabriel's Theorem in the Dynkin type A case. The proof uses the fact that the quivers of the cluster-tilted algebras of Dynkin type A have a particularly nice description in terms of triangulation of regular polygons.

This work is a study of p-permutation modules for the group algebra of S2p and also for the group algebras of some of its subgroups over an algebraically closed field of odd characteristic

The countable generic partial order (P, <) is defined to be the Fraisse limit of the class of finite partially ordered sets. It occurred as part of Schmerl's classification of countable homogeneous partial orders (see [5]). In this thesis I study its automorphisms.

In Part I of the thesis I investigate the invariants of the divided powers algebra, Γ(<i>V</i>) under the action of the General and Special Linear groups.  In Chapter 2 I compare the additive structure of the invariants of Γ(<i>V</i>) with that of the invariants of the polynomial algebra.  I show that these are not isomorphic as vector spaces (and <i>a fortiori</i> not isomorphic as algebras) except in two cases – when dim(<i>V</i>) = 2 and <i>p</i> = 2 or 3.  This chapter also includes some dimension counting arguments most notably the case dim(<i>V</i>) = 2, <i>p</i> = 2 in Section 2.7.  These dimension counts are useful both as techniques in their own right and because they give explicit calculations which are useful in the following chapter. In Chapter 3 I describe the algebra structure of the invariant algebra in the two cases, dim(<i>V</i>) = 2 and <i>p</i> = 2 or 3 in sections 3.4 and 3.6 respectively.  In addition I describe the algebra structure of the invariants of Γ(<i>V</i>) under some important subgroups of <i>GL</i>(<i>V</i>) – the transvections, the symmetric subgroup and the multiplicative subgroup.  I give complete results for the transvections and F<i>_p</i>^x_..  For the symmetric invariants I correct a result of Joel Segal and give a complete description of Γ<i>_p</i>(<i>V_p</i>)^Σ<i>p</i>_..  The method used in the description of Γ(<i>V</i>)<i>^SL</i>^3(<i>V</i>2)has potential to be extended to other cases. In Part II of the thesis I compare two different methods of defining some kind of Steenrod Operations in integral cohomology.  John Hubbuck’s K-theory squares are defined on any space homotopic to a finite CW-complex with no 2-torsion.  Reg Wood’s differential operator squares are defined only on polynomial algebras.  The question is whether considered on a suitable space these operations would be equivalent.  I show that they are essentially incompatible.

A Hopf bundle framework is constructed within Cn, in terms of which general paths on Cn \ {0} are viewed and analyzed. The resulting hierarchy of spaces is addressed both theoretically and numerically, and the consequences for numerics and applications are investigated through a wide range of numerical experiments. The geometric reframing of Cn in this way - in terms of an intrinsic fibre bundle - allows for the introduction of bundle-theoretic quantities in a general dynamical setting. The roles of the various structural elements of the bundle are explored, including horizontal and vertical subspaces, parallel translation and connections. These concepts lead naturally to the association of a unique geometric phase with each path on Cn \ {0}. This phase quantity is interpreted as a measure of the spinning in the S1 fibre of the Hopf bundle induced by paths on Cn \ {0}, relative to a given connection, and is shown to be an important quantity. The implications of adopting this bundle viewpoint are investigated in two specific contexts. The first is the case of the lowest-dimensional Hopf bundle, S1 &rarr; S3 &rarr; S2. Here the quaternionic matrices are used to develop a simplified, geometrically intuitive formulation of the bundle structure, and a reduced expression for the phase is used to compute numerical phase results in three example systems. The second is the case where paths in Cn \ {0} are generated by solutions to a particular class of parameter-dependent first-order ODEs. This establishes a direct link between the dynamical characteristics of such systems and the underlying bundle geometry. A variety of systems are examined and numerical phase results compiled. The numerics reveal an important correlation between the spectral properties of the path-generating ODEs and the resultant geometric phase change values. The details of this observed link are recorded in a conjecture.

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La thèse est consacrée à la construction et à l'étude des espaces de modules des connexions holomorphes algébroïdes de Lie sont étudiés.On commence par une classification des faisceaux d'algèbres filtrées quasi-polynômiales sur une variété complexe lisse projective en termes d'algébroïdes de Lie holomorphes et de leurs classes de cohomologie. Cela permet de construire les espaces de modules de connexions holomorphes agébroïdes de Lie par le formalisme des Lambda-modules de Simpson.Par ailleurs, on étudie la théorie des déformations de telles connexions, et on calcule le germe de leur espace de modules dans le cas de rang deux, lorsque la variété de base est une courbe. / The thesis is concerned with the consturction and the sudy of moduli spaces of holomorphic Lie algebroid connections. It provides a classification of sheaves of almost polynomials filtered algebras on a smooth projective complex variety in terms of holomorphic Lie algebroids and their cohomology classes. This permits to build moduli spaces of holomorphic Lie agebroid connections via Simpson's formalism of Lambda-modules. Furthermore, the deformation theory of such connections is suried, and the germ of their moduli spaces in the rank two case is computed when the base variety is a curve.

Στο Κεφάλαιο 1 δίνουμε τις έννοιες της ομοιομορφίας και του ομοιόμορφου χώρου. Προσδιορίζεται η σχέση ομοιόμορφων και τοπολογικών χώρων. Αποδεικνύεται ότι ο μονοσήμαντα ορισμένος τοπολογικός χώρος που προσδιορίζει ένας ομοιόμορφος χώρος είναι Tychonoff και ότι κάθε χώρος Tychonoff προσδιορίζεται (όχι μονοσήμαντα) από έναν ομοιόμορφο χώρο. Μελετώνται ιδιότητες των ομοιόμορφων χώρων και παραθέτονται παραδείγματα αυτών. Στο Κεφάλαιο 2 ορίζονται και μελετώνται οι ομοιομόρφως συνεχείς απεικονίσεις και διάφορες ιδιότητες των ομοιόμορφων χώρων. Στο κεφάλαιο 3 ορίζονται και μελετώνται ολικά φραγμένοι, πλήρεις και συμπαγείς ομοιόμορφοι χώροι. Στο Κεφάλαιο 4 δίνονται εφαρμογές των ομοιόμορφων χώρων σε χώρους συναρτήσεων. / -