Lattices and automorphisms of compact complex manifolds

This work makes use of well-known integral lattices to construct complex algebraic varieties reflecting properties of the lattices. In particular the automorphism groups of the lattices are closely related to the symmetries of varieties. The constructions are to two types: generalised Kummer manifolds and toric varieties. In both cases the examples are of the most interest. A generalised Kummer manifold is the resolution of the quotient of a complex torus by some finite group G. A description of the construction for certain cyclic groups G by given in terms of holomorphic surgery of disc bundles. The action of the automorphism groups is given explicitly. The most important example is a compact complex 12-dimensinoal manifold associated to the Leech lattice admitting an action of the finite simple Suzuki group. All these generalised Kummer manifolds are shown to be simply connected. Toric varieties are associated to certain decompositions of Rn into convex cones. The automorphism groups of those associated to Weyl group decompositions of Rn are calculated. These are used to construct 24-dimensional singular varieties from some Neimeier lattices. Their symmetries are extensions of Mathieu groups and their singularities closely related to the Golay codes

Incidence geometry from an algebraic graph theory point of view

The goal of this thesis is to apply techniques from algebraic graph theory to finite incidence geometry. The incidence geometries under consideration include projective spaces, polar spaces and near polygons. These geometries give rise to one or more graphs. By use of eigenvalue techniques, we obtain results on these graphs and on their substructures that are regular or extremal in some sense. The first chapter introduces the basic notions of geometries, such as projective and polar spaces. In the second chapter, we introduce the necessary concepts from algebraic graph theory, such as association schemes and distance-regular graphs, and the main techniques, including the fundamental contributions by Delsarte. Chapter 3 deals with the Grassmann association schemes, or more geometrically: with the projective geometries. Several examples of interesting subsets are given, and we can easily derive completely combinatorial properties of them. Chapter 4 discusses the association schemes from classical finite polar spaces. One of the main applications is obtaining bounds for the size of substructures known as partial m- systems. In one specific case, where the partial m-systems are partial spreads in the polar space H(2d − 1, q^2) with d odd, the bound is new and even tight. A variant of the famous Erdős-Ko-Rado problem is considered in Chapter 5, where we study sets of pairwise non-trivially intersecting maximal totally isotropic subspaces in polar spaces. A combination of geometric and algebraic techniques is used to obtain a classification of such sets of maximum size, except for one specific polar space, namely H(2d − 1, q^2) for odd rank d ≥ 5. Near polygons, including generalized polygons and dual polar spaces, are studied in the last chapter. Several results on substructures in these geometries are given. An inequality of Higman on the parameters of generalized quadrangles is generalized. Finally, it is proved that in a specific dual polar space, a highly regular substructure would yield a distance- regular graph, generalizing a result on hemisystems. The appendix consists of an alternative proof for one of the main results in the thesis, a list of open problems and a summary in Dutch

Probabilistic Arguments in Mathematics

This thesis addresses a question that emerges naturally from some observations about contemporary mathematical practice. Firstly, mathematicians always demand proof for the acceptance of new results. Secondly, the ability of mathematicians to tell if a discourse gives expression to a proof is less than perfect, and the computers they use are subject to a variety of hardware and software failures. So false results are sometimes accepted, despite insistence on proof. Thirdly, over the past few decades, researchers have also developed a variety of methods that are probabilistic in nature. Even if carried out perfectly, these procedures only yield a conclusion that is very likely to be true. In some cases, these chances of error are precisely specifiable and can be made as small as desired. The likelihood of an error arising from the inherently uncertain nature of these probabilistic algorithms can therefore be made vanishingly small in comparison to the chances of an error arising when implementing an equivalent deductive algorithm. Moreover, the structure of probabilistic algorithms tends to minimise these Implementation Errors too. So overall, probabilistic methods are sometimes more reliable than deductive ones. This invites the question: ‘Are mathematicians rational in continuing to reject these probabilistic methods as a means of establishing mathematical claims?