Nowadays decision centres are required to make choices in complex and evolving environments, described through multiple and interdependent processes with many associated measurements. The objective of a real time decision making centre is to agree to a sequence of efficacious countermeasures. To achieve this it is usually necessary to integrate opinions and information from an often diverse set of stakeholders, articulating several competing objectives and knowledge over different domains of expertise. A collection of decision support systems can enhance such an integration, not only ensuring that all relevant evidence systematically informs policy making, but also encouraging the decision centre to exhibit an underlying consistency across all its components and to address the problem as a whole. In this thesis we develop a formal framework, extending standard Bayesian methodology, enabling the judgements and the models of groups of experts to be coherently aggregated in a unique entity. We discuss when and how it is possible to do so and the conditions the group needs to agree upon. We call this framework integrating decision support system. We then develop a variety of methodologies to enhance such an integration, enabling integrating decision support systems to be feasibly used in practice.

Let R be a regular ring such that 2 is invertible. We construct a spectral sequence converging to the hermitian K-theory, alias the Grothendieck-Witt theory, of R. In particular, we construct a tower for the hermitian K-groups in even shifts, whose terms are given by the hermitian K-theory of automorphisms. The spectral sequence arises as the homotopy spectral sequence of this tower and is analogous to Grayson’s version of the motivic spectral sequence [Gra95]. Further, we construct similar towers for the hermitian K-theory in odd shifts if R is a field of characteristic different from 2. We show by a counter example that the arising spectral sequence does not behave as desired. We proceed by proposing an alternative version for the tower and verify its correctness in weight 1. Finally we give a geometric representation of the (hermitian) K-theory of automorphisms in terms of the general linear group, the orthogonal group, or in terms of e-symmetric matrices, respectively. The K-theory of automorphisms can be identified with motivic cohomology if R is local and of finite type over a field. Therefore the hermitian K-theory of automorphisms as presented in this thesis is a candidate for the analogue of motivic cohomology in the hermitian world.

Estimating multiple geometric shapes such as tracks or surfaces creates significant mathematical challenges particularly in the presence of unknown data association. In particular, problems of this type have two major challenges. The first is typically the object of interest is infinite dimensional whilst data is finite dimensional. As a result the inverse problem is ill-posed without regularization. The second is the data association makes the likelihood function highly oscillatory. The focus of this thesis is on techniques to validate approaches to estimating problems in geometric statistical inference. We use convergence of the large data limit as an indicator of robustness of the methodology. One particular advantage of our approach is that we can prove convergence under modest conditions on the data generating process. This allows one to apply the theory where very little is known about the data. This indicates a robustness in applications to real world problems. The results of this thesis therefore concern the asymptotics for a selection of statistical inference problems. We construct our estimates as the minimizer of an appropriate functional and look at what happens in the large data limit. In each case we will show our estimates converge to a minimizer of a limiting functional. In certain cases we also add rates of convergence. The emphasis is on problems which contain a data association or classification component. More precisely we study a generalized version of the k-means method which is suitable for estimating multiple trajectories from unlabeled data which combines data association with spline smoothing. Another problem considered is a graphical approach to estimating the labeling of data points. Our approach uses minimizers of the Ginzburg-Landau functional on a suitably defined graph. In order to study these problems we use variational techniques and in particular I-convergence. This is the natural framework to use for studying sequences of minimization problems. A key advantage of this approach is that it allows us to deal with infinite dimensional and highly oscillatory functionals.

Chapter 1 we use a Poisson stochastic measure to establish a method of localizing, and a change of chart formula for, a class of stochastic differential equations with discontinuous sample paths. This is based on Gikhman and Skorohod [4]. In Chapter 2 we use essentially the method of Elworthy [2], to construct a unique, maximal solution to a stochastic differential equation defined on a manifold M. Chapter 3 establishes some properties of solutions of the equation. In particular if M is compact, then the solutions have infinite explosion time. We evaluate the infinitesimal generator of the process. By defining stochastic development of a-stable processes on the tangent space, we produce a process on the manifold which, as is shown in Section 6, is not a-stable on M.

This thesis addresses three main topics related to the practical problems of modelling the spread of nuclear material after an accidental release. The first topic deals with the issue of how qualitative information (expert jUdgement) about the development of the emission of contamination after an accident can be coded as a Dynamic Linear Model (DLM). An illustration is given of the subsequent adaptation of the expert judgement in response to the incoming data. Moreover, the height of the release at the source can be a key parameter in the subsequent dispersal. We addressed uncertainty on the release height using the Multi-Process Models framework. That is we included several models in our analysis, each with a different release height. The Bayesian methodology uses probabilities representing their relative likelihood to weight these and updates the probabilities in the light of monitoring data. A brief illustration of testing the updating algorithm on simulated contamination readings is provided. The second topic concerns the demands of computational efficiency. We show how the Bayesian propagation algorithms on a dynamic junction tree of cliques of variables (representing a high dimensional Gaussian process), as provided by Smith et al. (1995), can be generalised to incorporate the case when data may destroy neat dependencies (i.e. when observations are taken under more than one clique). Here we introduce two classes of new operators: exact and non-exact (approximations) which act on this high dimensional Gaussian process, modifying its junction tree by another tree which allows quicker probability propagation. We also develop fast algorithms which can be defined by approximating Gaussian systems by cutting edges on junctions. The appropriateness ofthe approximations is based on the Kulback-Leibler/Hellinger distances. Some of these new operators and algorithms have been implemented and coded. Preliminary tests on these algorithms were carried out using arbitrary data, and the system proved to be highly efficient in terms of P.C. user time. The third topic concentrates on generalisations from a Gaussian process. It proposes, as a good approximation, an adaptation of the Dynamic Generalised Linear Models (DGLMs) of West, Harrison, and Migon (1985) for updating algorithms on a dynamic junction tree. The Hellinger distance is used to check the accuracy of the dynamic approximation. The analysis of these topics involves a review and extension of some useful theory and results on Bayesian forecasting and dynamic models, graphical modelling, and information divergence.

A flip is a birational map of 3-folds X- ---> X+ which is an isomorphism away from curves C- c X- and C+ c X+ and does not extend across these curves. Flips are the primary object of study of this thesis. I discuss their formal definition and history in Chapter 1. Flips are well known in toric geometry. In Chapter 2, I calculate how the numbers K3 and χ(nK) differ between X- and X+ for toric flips. These numbers are also related in a primary way by Riemann-Roch theorems but I keep that quiet until Chapter 5. In Chapter 3, I describe a technique, which I learned from Miles Reid, for constructing a flip as C* quotients of a local variety 0 E A, taken in different ways. The codimension of my title refers to the minimal embedding dimension of 0 E A. The case of codimension 0 turns out to be exactly the case of toric geometry as studied in Chapter 2. The main result of Chapter 3 classifies the cases when A c C5 is a singular hypersurface, that is, when A defines a flip in codimension 1. Chapters 4 and 5 concern themselves with computing new examples of flips in higher codimension and studying changes in general flips. I indicate one benefit of knowing how these changes work. The main results of Chapters 2 and 3 have been circulated informally as [2] and [3] respectively.

(1) We define normality for fuzzy topological spaces, define a fuzzy unit interval, and prove a Urysohn type lemma. (2) We define uniformities on fuzzy lattices, and characterise uniformizability in terms of complete regularity. (3) We define the product of a collection of fuzzy topological spaces. We define compactness and connectedness, and show that the product is compact (connected) iff each factor space is. (4) We place normality and complete regularity within a coherent hierarchy of separation and regularity axioms. We prove the usual implications, and the usual theorems about compactness and products. (5) We give alternative definitions of uniformities and pseudometrics, and show a compact R1 space has a unique uniformity.

The central theme of this thesis is the behavior of the value function of general optimal stopping problems under a stochastic volatility model when varying the volatility dynamics. We first use a combination of time-change and coupling techniques to show regularity properties of the value function. We consider a large class of terminal payoffs: when the first component of the model is a stochastic differential equation without drift we allow for general measurable functions, and when it has a drift we impose a mild condition which includes possibly unbounded and discontinuous functions. We also consider a running cost which can be any non-negative and bounded Borel function. Moreover, we derive the solution of a zero-sum game of stopping and control, which arises when considering some parameter uncertainty in the volatility dynamics. In both finite and infinite horizon, we exhibit the existence of a saddle point using stochastic control and martingale arguments as well as the probabilistic representation of solutions to free-boundary problems. Overall, our approach in mainly theoretical, however we impose only verifiable conditions. We then discuss some examples arising in American option pricing where our results are applicable. In particular, we are able to compare American option prices under different volatility models in a variety of settings and we establish that the optimal exercise boundary for the associated option is a monotone function of the volatility.

Let G C GL3(C) be the group of type 1/r(1, a, r-a) with a coprime to r. For such G, the quotient variety X = C3/G is not Gorenstein and has a terminal singularity. The singular variety X has the economic resolution which is "close to being crepant". In this paper, we prove that the economic resolution of the quotient variety X = C3/G is isomorphic to the birational component of a moduli space of Θ-stable McKay quiver representations for a suitable GIT parameter Θ. Moreover, we conjecture that the moduli space of Θ-stable McKay quiver representations is irreducible, and prove this for a = 2 and in a number of special examples.

We introduce, and analyze in terms of convergence rates of transition kernels, a continuous-time Markov chain approximation to Lévy processes. A full fluctuation theory for what are right-continuous random walks embedded into continuous-time as compound Poisson processes, is provided. These results are applied to obtaining a general algorithm for the calculation of the scale functions of a spectrally negative Lévy process. In a related result, the class of Lévy processes having non-random overshoots is precisely characterized.