Doctor of Philosophy / Department of Mathematics / Yan S. Soibelman / Recall the classical Riemann-Roch theorem for curves: Given a smooth projective complex curve and two holomorphic vector bundles E, F on it, the Euler can be computed in terms of the ranks and the degrees of the vector bundles. Remarkably, there are a number of similarly looking formulas in algebra. The simplest example is the Ringel formula in the theory of quivers. It expresses the Euler form of two finite-dimensional representations of a quiver algebra in terms of a certain pairing of their dimension vectors. The existence of Riemann-Roch type formulas in these two settings is a consequence of a deeper similarity in the structure of the corresponding derived categories - those of sheaves on curves and of modules over quiver algebras. The thesis is devoted to a version of the Riemann-Roch formula for abstract derived categories. By the latter we understand the derived categories of differential graded (DG) categories. More specifically, we work with the categories of perfect modules over DG algebras. These are a simultaneous generalization of the derived categories of modules over associative algebras and the derived categories of schemes. Given an arbitrary DG algebra A, satisfying a certain finiteness condition, we define and explicitly describe a canonical pairing on its Hochschild homology. Then we give an explicit formula for the Euler character of an arbitrary perfect A-module, the character is an element of the Hochschild homology of A. In this setting, our noncommutative Riemann-Roch formula expresses the Euler characteristic of the Hom-complex between any two perfect A-modules in terms of the pairing of their Euler characters. One of the main applications of our results is a theorem that the aforementioned pairing on the Hochschild homology is non-degenerate when the DG algebra satisfies a smoothness condition. This theorem implies a special case of the well-known noncommutative Hodge-to-de Rham degeneration conjecture. Another application is related to mathematical physics: We explicitly construct an open-closed topological field theory from an arbitrary Frobenius algebra and then, following ideas of physicists, interpret the noncommutative Riemann-Roch formula as a special case of the so-called topological Cardy condition.

Snooker scores from two complete rounds of graded teams competition are analysed to detect differences in scores which can be attributed to various external factors, in order to quantify the factors relevant to a player's score in a game of snooker. Such factors are assessed subjectively at present. While each factor examined is found to be significant in one or more of the various grades, it is only in the highest grade that a clear pattern to matches can be identified.

There is no abstract available for this dissertation.

Piezoelectric actuators have been developed in various forms ranging from discrete layered composites to functionally graded composites. These composite actuators are usually made up of differentially poled piezoelectric ceramics. This study presents analyses of thermo-electro-mechanical response of piezoelectric actuators having combinations of metal and ceramic constituents with through thickness gradual variations of the metal and ceramic compositions. This is done in order to achieve better performance. The piezoelectric ceramic constituent allows for electro-mechanical coupling response and higher resistance to elevated temperatures while the metal constituent provides more ductile composites. The gradual variation in the ceramic and metal composition helps to avoid high stress concentrations at the layer interfaces in composites. A functionally graded composite is analyzed with discrete layers of piezoelectric ceramic/metal composite. Each layer in the functionally graded composite has a fixed ceramic/metal composition. The governing equation for such a piezoelectric functionally composite beam is presented based on a multi-layer Euler-Bernoulli beam model and the overall displacement response of the beam under thermal, mechanical and electrical stimuli is predicted. The variation of this response is studied with respect to functional grading parameter, number of layers, thermal and electrical and mechanical stimuli applied. It is found that the displacement due to thermal and mechanical effects can be mitigated to some extent by the application of an electric field. It is also observed that layers of varying thickness may be assumed to model the functional grading more accurately i.e. use thinner layers where the grading changes rapidly and thicker layers where the grading changes gradually. In addition to the above parametric studies, the change in the material properties with temperature is also studied. It is found that the temperature-dependent material parameters are important when the actuators are subjected to elevated temperatures.

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Published Article / This paper looks at the development of composite materials in layered manufacturing. It is known that Rapid Prototyping (RP) using a single material compares poorly with other conventional manufacturing processes when making parts from similar materials. For example, injection moulded parts are over 30% stronger than RP fabricated parts of the same material. The incorporation of secondary materials can result in a composite that can improve this situation. This paper will discuss different composites that are commercially available as well as some into which research is being conducted. An advantage of RP is that composites do not have to be manufactured in a homogeneous manner. Functionally graded parts may be fabricated where reinforcing material can be added in appropriate locations and in required orientations.

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The Khovanov{Lauda{Rouquier algebras Rn are a relatively new family of Z-graded algebras. Their cyclotomic quotients R n are intimately connected to a smaller family of algebras, the cyclotomic Hecke algebras H n of type A, via Brundan and Kleshchev's Graded Isomorphism Theorem. The study of representation theory of H n is well developed, partly inspired by the remaining open questions about the modular representations of the symmetric group Sn. There is a profound interplay between the representations for Sn and combinatorics, whereby each irreducible representation in characteristic zero can be realised as a Specht module whose basis is constructed from combinatorial objects. For R n , we can similarly construct their representations as analogous Specht modules in a combinatorial fashion. Many results can be lifted through the Graded Isomorphism Theorem from the symmetric group algebras, and more so from H n , to the cyclotomic Khovanov{Lauda{Rouquier algebras, providing a foundation for the representation theory of R n . Following the introduction of R n , Brundan, Kleshchev and Wang discovered that Specht modules over R n have Z-graded bases, giving rise to the study of graded Specht modules. In this thesis we solely study graded Specht modules and their irreducible quotients for R n . One of the main problems in graded representation theory of R n , the Graded Decomposition Number Problem, is to determine the graded multiplicities of graded irreducible R n -modules arising as graded composition factors of graded Specht modules. We rst consider R n in level one, which is isomorphic to the Iwahori{Hecke algebra of type A, and research graded Specht modules labelled by hook partitions in this context. In quantum characteristic two, we extend to R n a result of Murphy for the symmetric groups, determining graded ltrations of Specht modules labelled by hook partitions, whose factors appear as Specht modules labelled by two-part partitions. In quantum characteristic at least three, we determine an analogous R n -version of Peel's Theorem for the symmetric groups, providing an alternative approach to Chuang, Miyachi and Tan. We then study graded Specht modules labelled by hook bipartitions for R n in level two, which is isomorphic to the Iwahori{Hecke algebra of type B. In quantum characterisitic at least three, we completely determine the composition factors of Specht modules labelled by hook bipartitions for R n , together with their graded analogues.

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