We study the SU(3) AVE graphs, which appear in the classification of modular in variant partition functions from numerous viewpoints, including determination of their Boltzmann weights, representations of Hecke algebras, a new notion of A2 planar algebras and their modules, various Hilbert series of dimensions and spectral measures, and the K-theory of associated Cuntz-Krieger algebras. We compute the K-theory of the of the Cuntz-Krieger algebras associated to the SU(3) AVE graphs. We compute the numerical values of the Ocneanu cells, and consequently representations of the Hecke algebra, for the AVE graphs. Some such representations have appeared in the literature and we compare our results. We use these cells to define an SU(3) analogue of the Goodman-de la Harpe-Jones construction of a subfactor, where we embed the j42-Temperley-Lieb algebra in an AF path-algebra of the SU(3) AVE graphs. Using this construction, we realize all SU(3) modular invariants by subfactors previously announced by Ocneanu. We give a diagrammatic representation of the i42-Temperley-Lieb algebra, and show that it is isomorphic to Wenzl's representation of a Hecke algebra. Generalizing Jones's notion of a planar algebra, we construct an 42-planar algebra which captures the structure contained in the SU(3) AVE subfactors. We show that the subfactor for an AVE graph with a flat connection has a description as a flat >12-planar algebra. We introduce the notion of modules over an 42-planar algebra, and describe certain irreducible Hilbert A2- Temperley-Lieb-modules. A partial decomposition of the ,42-planar algebras for the AVE graphs is achieved. We compare various Hilbert series of dimensions associated to ADE models for SU(2), and the Hilbert series of certain Calabi-Yau algebras of dimension 3. We also consider spectral measures for the ADE graphs and generalize to SU(3), and in particular obtain spectral measures for the infinite SU(3) graphs.

The major part of my thesis is concerned with the size and structure of Sylow p-subgroups of a primitive permutation group. The results of Theorems 2.2 and 2.3 were suggested by similar results of Jordan, Manning, Waiss, and othera, about elements of order p in a primitive group. The following are the three main results: Theorem 2.1. If G is a transitive permutation group on a set Ω of degree n, and if P is a Sylow p-subgroup of G for some prime p dividing |G|, then the number of points of Ω fixed by P is less than ⁿ⁄₂. Theorem 2.2. Let G be a primitive permutation group on Ω of degree n = kp, where p is a prime, and such that G does not contain the alternating group A_n. Let P be a Sylow p-subgroup of G, and suppose that P has no orbits of length greater thin p. Then P has order p unless</p> <ol type=a> <li>|P| = 4 and G is PSL(2,5) permuting the 6 points or the 1-dimensional projective geometry PG(1,5), or</li> <li>|P| = 9 and G is the Mathieu group M₁₁ in its 3-transitive representation of degree 12.</li> </ol> <p>This result is due to L. Scott for the case in which G is not 2-transitive and my contribution is the 2-transitive case. Theorem 2.3. Let G be a 2-transitive permutation group on Ω of degree n = kp + f, for some prime p, such that G does not contain the alternating group A_n. Suppose that p divides |G| and that a Sylow p-subgroup P of G has k orbits of length p and f fixed points in Ω. Then P has order p unless f = 0. As the first application of these results we prove Theorem 7.1 below about 2-transitive groups of degree r² + 3r + 3, where r is a prime. This problem arose from a conjecture about transitive groups of prime degree, and work of Peter Neumann and Tom McDonough. Theorem 7.1. If G is a 2-transitive permutation group on Ω of degree n = r² + 3r + 3, where r is a prime greater than 3, and such that r divides |G|, then either G contains the alternating group A_n, or r is of the form 2^m - 1, a Mersenne prime, for some odd prime m, and G is such that PSL(3,2^m) ≤ G ≤ PΓL(3,2^m). Next we turn to 2-transitive groups of degree p², where p is a prime. In looking at the case whore the Sylow p-subgroups are cyclic, the situation arose in which G had an indecomposable representation of degree less than ^|P|⁄₂. To deal with this, the next theorem, an extension of a result of Felt, was proved. Theorem 9.2. Let G be a finite group with a cyclic Sylow p-subgroup P of order p^k ≥ p², which is a T.I. set. Suppose that G is not p-soluble. Suppose that G has an indecomposable representation â„’ in a field K of characteristic p of degree d ≤ p^k, such that P is not contained in the kernel of â„’. Then â„’_p is indecomposable, C_G(P) = PxZ(G), and d ≥ ^{(p^k+1)}⁄₂. Finally there are some results about 2-transitive groups of degree p², following on from Wielendt's classification of the simply transitive groups: Theorem 12.3. If G is a 2-transitive group of degree p² and P is a Sylow p-subgroup of G, then either</p> <ol type=a> <li>|P| ≥ p⁴ and G contains A_p², for p ≥ 3, or</li> <li>|P| = p³ and G ≤ Aff(2,p), (and G has PSL(2,p) as a composition factor), or</li> <li>|P| = 3³ and G is PΓL(2,8) of degree 9, or</li> <li>|P| = 2³ and G is S₄ of degree 4, or</li> <li>|P| = p².</li> </ol> <p>If G is primitive of degree p^k and its Sylow p-subgroups are cyclic, we use Theorem 9.2 to extend results of Neumann and Ito, (Theorem 14.2, and Corollary 14.3).

In this thesis we address several questions related to important conjectures in birational geometry. In the first two chapters we prove that it is possible to bound the number of minimal models of a smooth threefold of general type depending on the topology of the underlying complex manifold. Moreover, under some technical assumptions, we provide some explicit bounds and we explain the relationship with the effective version of the finite generation of the canonical ring. Then we prove the existence of rational curves on certain type of fibered Calabi-Yau manifolds. Finally, in the last chapter we move to birational geometry in positive characteristic and we prove the Base point free Theorem for a three dimensional log canonical pair over the algebraic closure of a finite field.

Let G be a complex non-exceptional simple algebraic group and g its Lie algebra. With every point x of the maximal torus T ʗ G we associate a highest weight module Mx over the Drinfeld-Jimbo quantum group Uq(g) and an equivariant quantization of the conjugacy class of x by operators in End(Mx). These equivariant quantizations are isomorphic for x lying on the same orbit of the Weyl group, and Mx support different exact representations of the same quantum conjugacy class. This recovers all quantizations of conjugacy classes constructed before, via special x, and also completes the family of conjugacy classes by constructing an equivariant quantization of “borderline" Levi conjugacy classes of the complex orthogonal group SO(N), whose stabilizer contains a Cartesian factor SO(2) SO(P), 1 6 P < N, P Ξ N mod 2. To achieve this, generators of the Mickelsson algebra Zq(g; g’), where g’ ʗ g is the Lie subalgebra of rank rkg’ = rkg-1 of the same type, were explicitly constructed in terms of Chevalley generators via the R-matrix of Uq(g).

This thesis concerns the representation theory of diagram algebras and related problems. In particular, we consider subalgebras and generalisations of the partition algebra. We study the d-tonal partition algebra and the planar d-tonal partition algebra. Regarding the d-tonal partition algebra, a complete description of the J -classes of the underlying monoid of this algebra is obtained. Furthermore, the structure of the poset of J -classes of the d-tonal partition monoid is also studied and numerous combinatorial results are presented. We observe a connection between canonical elements of the d-tonal partition monoids and some combinatorial objects which describe certain types of hydrocarbons, by using the alcove system of some reflection groups. We show that the planar d-tonal partition algebra is quasi-hereditary and generically semisimple. The standard modules of the planar d-tonal partition algebra are explicitly constructed, and the restriction rules for the standard modules are also given. The planar 2-tonal partition algebra is closely related to the two coloured Fuss-Catalan algebra. We use this relation to transfer information from one side to the other. For example, we obtain a presentation of the 2-tonal partition algebra by generators and relations. Furthermore, we present a necessary and sufficient condition for semisimplicity of the two colour Fuss-Catalan algebra, under certain known restrictions.

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In this thesis we will be concerned with some questions regarding involutions on dual and predual spaces of certain algebras arising from locally compact quantum groups. In particular we have the $L^1(\G)$ predual of a von Neumann algebraic quantum group $(L^\infty(\G), \Delta)$. This is a Banach algebra (where the product is given by the pre-adjoint of the coproduct $\Delta$), however in general we cannot make this into a Banach $*$-algebra in such a way that the regular representation is a $*$-homomorphism. We can however find a dense $*$-subalgebra $L^1_\sharp(\G)$ that satisfies this property and is a Banach algebra under a new norm. This was originally considered by Kustermans and Vaes when defining the universal C$^*$-algebraic quantum group, however little else has been studied regarding this algebra in general. In this thesis we study the $L^1_\sharp$-algebra of a locally compact quantum group in this thesis. In particular we show how this has a (not necessarily unique) operator space structure such that this forms a completely contractive Banach algebra, we study some properties for compact quantum groups, we study the object for the compact quantum group $\mathrm{SU}_q(2)$ and we study the operator biprojectivity of the $L^1_\sharp$-algebra. In addition we also briefly study some related properties of $C_0(\G)^*$ and its $*$-subalgebra ${C_0(\G)^*}_\sharp$.

We study the derived category of a complete intersection X of bilinear divisors in the orbifold Sym^2 P(V). Our results are in the spirit of Kuznetsov’s theory of homological projective duality, and we describe a homological projective duality relation between Sym^2 P(V) and a category of modules over a sheaf of Clifford algebras on P(Sym^2 V^\vee). The proof follows a recently developed strategy combining variation of GIT stability and categories of global matrix factorisations. We begin by translating D^b(X) into a derived category of factorisations on an LG model, and then apply VGIT to obtain a birational LG model. Finally, we interpret the derived factorisation category of the new LG model as a Clifford module category. In some cases we can compute this Clifford module category as the derived category of a variety. As a corollary we get a new proof of a result of Hosono and Takagi, which says that a certain pair of nonbirational Calabi–Yau 3-folds have equivalent derived categories.

This thesis is concerned with the mathematical formulations of higher gauge theory. Firstly, we develop a complete description of principal 2-bundles with string 2-group model of Schommer-Pries, which is obtained by defining principal smooth 2-group bundles as internal functors in the weak 2-category Bibun of Lie groupoids, right principal smooth bibundles and bibundle maps. Furthermore, this formalism allows us to construct the known string Lie 2-algebra by differentiating this model of the string 2-group. Generalizing the differentiation process, we provide Maurer- Cartan forms leading us to higher non-abelian Deligne cohomology, encoding the kinematical data of higher gauge theory together with their (finite) gauge symmetries. Secondly, we prove the non-abelian Poincare lemma in higher gauge theory in two different ways. That is, we show that every flat local connective structure in strict principal 2-bundles is gauge trivial. The first proof is based on the result by Jacobowitz, which explains solvability conditions for equations of differential forms. The second is an extension of a proof by T. Voronov and yields the explicit gauge parameters connecting a flat local connective structure to the trivial one. Finally, we develop a method that shows how higher flatness appears as a necessary integrability condition of a linear system by translating the usual matrix product into categorised settings. Moreover, we comment how this notion can be also generalized to the case of higher principal bundles with connective structures.