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Surface pasting is an interactive hierarchical modelling technique used to construct surfaces with varying levels of local detail. The concept is similar to that of the physical process of modelling with clay, where features are placed on to a base surface and attached by a smooth join obtained by adjusting the feature. Cylindrical surface pasting extends this modelling paradigm by allowing for two base surfaces to be joined smoothly via a blending cylinder, as in attaching a clay head to the body using a neck. Unfortunately, computer-based pasting involves approximations that can cause cracks to appear in the composite surface. In particular this occurs when the pasted feature boundary does not lie exactly over the user-specified pasting region on the base surface. Determining pasted locations for the feature boundary control points that give a close to exact join is non-trivial, especially in the case of cylinders as their control points can not be defined to lie on their closed curve boundary. I propose and compare six simple methods for positioning a feature cylinder's control points such that the join boundary discontinuities are minimized. The methods considered are all algorithmically simple alternatives having low computational costs. While the results demonstrate an order of magnitude quality improvement for some methods on a convex-only curved base, as the complexity of the base surface increases, all the methods show similar performance. Although unexpected, it turns out that a simple mapping of the control points directly onto the pasting closed curve given on the base surface offers a reasonable cylindrical boundary pasting technique.

Surface pasting is an interactive hierarchical modelling technique used to construct surfaces with varying levels of local detail. The concept is similar to that of the physical process of modelling with clay, where features are placed on to a base surface and attached by a smooth join obtained by adjusting the feature. Cylindrical surface pasting extends this modelling paradigm by allowing for two base surfaces to be joined smoothly via a blending cylinder, as in attaching a clay head to the body using a neck. Unfortunately, computer-based pasting involves approximations that can cause cracks to appear in the composite surface. In particular this occurs when the pasted feature boundary does not lie exactly over the user-specified pasting region on the base surface. Determining pasted locations for the feature boundary control points that give a close to exact join is non-trivial, especially in the case of cylinders as their control points can not be defined to lie on their closed curve boundary. I propose and compare six simple methods for positioning a feature cylinder's control points such that the join boundary discontinuities are minimized. The methods considered are all algorithmically simple alternatives having low computational costs. While the results demonstrate an order of magnitude quality improvement for some methods on a convex-only curved base, as the complexity of the base surface increases, all the methods show similar performance. Although unexpected, it turns out that a simple mapping of the control points directly onto the pasting closed curve given on the base surface offers a reasonable cylindrical boundary pasting technique.

The goal of this thesis is to define and study the Fourier algebra A(G) of a locally compact groupoid G. If G is a locally compact group, its Fourier-Stieltjes algebra B(G) and its Fourier algebra A(G) were defined by Eymard in 1964. Since then, a rich theory has been developed. For the groupoid case, the algebras B(G) and A(G) have been studied by Ramsay and Walter (borelian case, 1997), Renault (measurable case, 1997) and Paterson (locally compact case, 2004). In this work, we present a new definition of A(G) in the locally compact case, specially well suited for studying locally trivial groupoids. Let G be a locally compact proper groupoid. Following the group case, in order to define A(G), we consider the closure under certain norm of the span of the left regular G-Hilbert bundle coefficients. With the norm mentioned above, the space A(G) is a commutative Banach algebra of continuous functions of G vanishing at infinity. Moreover, A(G) separates points and it is also a B(G)-bimodule. If, in addition, G is compact, then B(G) and A(G) coincide. For a locally trivial groupoid G we present an easier to handle definition of A(G) that involves "trivializing" the left regular bundle. The main result of our work is a decomposition of A(G), valid for transitive, locally trivial groupoids with a "nice" Haar system. The condition we require the Haar system to satisfy is to be compatible with the Haar measure of the isotropy group at a fixed unit u. If the groupoid is transitive, locally trivial and unimodular, such a Haar system always can be constructed. For such groupoids, our theorem states that A(G) is isomorphic to the Haagerup tensor product of the space of continuous functions on Gu vanishing at infinity, times the Fourier algebra of the isotropy group at u, times space of continuous functions on Gu vanishing at infinity. Here Gu denotes the elements of the groupoid that have range u. This decomposition provides an operator space structure for A(G) and makes this space a completely contractive Banach algebra. If the locally trivial groupoid G has more than one transitive component, that we denote Gi, since these components are also topological components, there is a correspondence between G-Hilbert bundles and families of Gi-Hilbert bundles. Thanks to this correspondence, the Fourier-Stieltjes and Fourier algebra of G can be written as sums of the algebras of the Gi components. The theory of operator spaces is the main tool used in our work. In particular, the many properties of the Haagerup tensor product are of vital importance. Our decomposition can be applied to (trivially) locally trivial groupoids of the form X times X and X times H times X, for a locally compact space X and a locally compact group H. It can also be applied to transformation group groupoids X times H arising from the action of a Lie group H on a locally compact space X and to the fundamental groupoid of a path-connected manifold.

When the tensor product of two irreducible representations contains multiple copies of some of its irreducible constituents, there is a problem of choosing specific copies: resolving the multiplicity. This is typically accomplished by some ad hoc method chosen primarily for convenience in labelling and calculations. This thesis addresses the possibility of making choices according to other criteria. One possible criterion is to choose copies for which the Clebsch-Gordan coefficients have a simple form. A method fulfilling this is introduced for the tensor product of three irreps of $su(2)$. This method is then extended to the tensor product of two irreps of $su(3)$. In both cases the method is shown to construct a full nested sequence of basis independent highest weight subspaces. Another possible criterion is to make choices which are intrinsic, independent of all choices of bases. This is investigated in the final part of the thesis with a basis independent method that applies to the tensor product of finite dimensional irreps of any semisimple Lie algebra over $\mathbb{C}$.

When the tensor product of two irreducible representations contains multiple copies of some of its irreducible constituents, there is a problem of choosing specific copies: resolving the multiplicity. This is typically accomplished by some ad hoc method chosen primarily for convenience in labelling and calculations. This thesis addresses the possibility of making choices according to other criteria. One possible criterion is to choose copies for which the Clebsch-Gordan coefficients have a simple form. A method fulfilling this is introduced for the tensor product of three irreps of $su(2)$. This method is then extended to the tensor product of two irreps of $su(3)$. In both cases the method is shown to construct a full nested sequence of basis independent highest weight subspaces. Another possible criterion is to make choices which are intrinsic, independent of all choices of bases. This is investigated in the final part of the thesis with a basis independent method that applies to the tensor product of finite dimensional irreps of any semisimple Lie algebra over $\mathbb{C}$.

The goal of this thesis is to define and study the Fourier algebra A(G) of a locally compact groupoid G. If G is a locally compact group, its Fourier-Stieltjes algebra B(G) and its Fourier algebra A(G) were defined by Eymard in 1964. Since then, a rich theory has been developed. For the groupoid case, the algebras B(G) and A(G) have been studied by Ramsay and Walter (borelian case, 1997), Renault (measurable case, 1997) and Paterson (locally compact case, 2004). In this work, we present a new definition of A(G) in the locally compact case, specially well suited for studying locally trivial groupoids. Let G be a locally compact proper groupoid. Following the group case, in order to define A(G), we consider the closure under certain norm of the span of the left regular G-Hilbert bundle coefficients. With the norm mentioned above, the space A(G) is a commutative Banach algebra of continuous functions of G vanishing at infinity. Moreover, A(G) separates points and it is also a B(G)-bimodule. If, in addition, G is compact, then B(G) and A(G) coincide. For a locally trivial groupoid G we present an easier to handle definition of A(G) that involves "trivializing" the left regular bundle. The main result of our work is a decomposition of A(G), valid for transitive, locally trivial groupoids with a "nice" Haar system. The condition we require the Haar system to satisfy is to be compatible with the Haar measure of the isotropy group at a fixed unit u. If the groupoid is transitive, locally trivial and unimodular, such a Haar system always can be constructed. For such groupoids, our theorem states that A(G) is isomorphic to the Haagerup tensor product of the space of continuous functions on Gu vanishing at infinity, times the Fourier algebra of the isotropy group at u, times space of continuous functions on Gu vanishing at infinity. Here Gu denotes the elements of the groupoid that have range u. This decomposition provides an operator space structure for A(G) and makes this space a completely contractive Banach algebra. If the locally trivial groupoid G has more than one transitive component, that we denote Gi, since these components are also topological components, there is a correspondence between G-Hilbert bundles and families of Gi-Hilbert bundles. Thanks to this correspondence, the Fourier-Stieltjes and Fourier algebra of G can be written as sums of the algebras of the Gi components. The theory of operator spaces is the main tool used in our work. In particular, the many properties of the Haagerup tensor product are of vital importance. Our decomposition can be applied to (trivially) locally trivial groupoids of the form X times X and X times H times X, for a locally compact space X and a locally compact group H. It can also be applied to transformation group groupoids X times H arising from the action of a Lie group H on a locally compact space X and to the fundamental groupoid of a path-connected manifold.

In this thesis, we study the representation theory of Lie colour algebras. Our strategy follows the work of G. Benkart, C. L. Shader and A. Ram in 1998, which is to use the Brauer algebras which appear as the commutant of the orthosymplectic Lie colour algebra when they act on a k-fold tensor product of the standard representation. We give a general combinatorial construction of highest weight vectors using tableaux, and compute characters of the irreducible summands in some borderline cases. Along the way, we prove the RSK-correspondence for tableaux and the PBW theorem for Lie colour algebras.

x, 81 p. : ill. A print copy of this thesis is available through the UO Libraries. Search the library catalog for the location and call number. / We give an exposition of Deligne's tensor category Rep(St) where t is not necessarily an integer. Thereafter, we give a complete description of the blocks in Rep(St) for arbitrary t. Finally, we use our result on blocks to decompose tensor products and classify tensor ideals in Rep(St). / Committee in charge: Victor Ostrik, Chairperson, Mathematics; Daniel Dugger, Member, Mathematics; Jonathan Brundan, Member, Mathematics; Alexander Kleshchev, Member, Mathematics; Michael Kellman, Outside Member, Chemistry

Exterior tensor products of elliptic operators on smooth manifolds and manifolds with conical singularities are used to obtain examples of elliptic operators on manifolds with edges that do not admit well-posed edge boundary and coboundary conditions.