DSpace Collection:

Automorphisms of free products of groups

Tue, 05 Feb 2013 00:00:00 GMT

Title: Automorphisms of free products of groups Authors: Griffin, James Thomas Abstract: The symmetric automorphism group of a free product is a group rich in algebraic structure and with strong links to geometric configuration spaces. In this thesis I describe in detail and for the first time the (co)homology of the symmetric automorphism groups. To this end I construct a classifying space for the Fouxe-Rabinovitch automorphism group, a large normal subgroup of the symmetric automorphism group. This classifying space is a moduli space of 'cactus products', each of which has the homotopy type of a wedge product of spaces. To study this space we build a combinatorial theory centred around 'diagonal complexes' which may be of independent interest. The diagonal complex associated to the cactus products consists of the set of forest posets, which in turn characterise the homology of the moduli spaces of cactus products. The machinery of diagonal complexes is then turned towards the symmetric automorphism groups of a graph product of groups. I also show that symmetric automorphisms may be determined by their categorical properties and that they are in particular characteristic of the free product functor. This goes some way to explain their occurence in a range of situations. The final chapter is devoted to a class of configuration spaces of Euclidean n-spheres embedded disjointly in (n+2)-space. When n = 1 this is the configuration space of unknotted, unlinked loops in 3-space, which has been well studied. We continue this work for higher n and find that the fundamental groups remain unchanged. We then consider the homology and the higher homotopy groups of the configuration spaces. Our last contribution is an epilogue which discusses the place of these groups in the wider field of mathematics. It is the functoriality which is important here and using this new-found emphasis we argue that there should exist a generalised version of the material from the final chapter which would apply to a far wider range of configuration spaces.

Collaborating queues: large service network and a limit order book

Mon, 08 Oct 2012 23:00:00 GMT

Title: Collaborating queues: large service network and a limit order book Authors: Yudovina, Elena Abstract: We analyse the steady-state behaviour of two different models with collaborating queues: that is, models in which "customers" can be served by many types of "servers", and "servers" can process many types of "customers". The first example is a large-scale service system, such as a call centre. Collaboration is the result of cross-trained staff attending to several different types of incoming calls. We first examine a load-balancing policy, which aims to keep servers in different pools equally busy. Although the policy behaves order-optimally over fixed time horizons, we show that the steady-state distribution may fail to be tight on the diffusion scale. That is, in a family of ever-larger networks whose arrival rates grow as O(r) (where r is a scaling parameter growing to infinity), the sequence of steady-state deviations from equilibrium scaled down by sqrt(r) is not tight. We then propose a different policy, for which we show that the sequence of invariant distributions is tight on the r^(1/2+epsilon) scale, for any epsilon > 0. For this policy we conjecture that tightness holds on the diffusion scale as well. The second example models a limit order book, a pricing mechanism for a single-commodity market in which buyers (respectively sellers) are prepared to wait for the price to drop (respectively rise). We analyse the behaviour of a simplified model, in which the arrival events are independent of each other and the state of the limit order book. The system can be represented by a queueing model, with "customers" and "servers" corresponding to bids and asks; the roles of customers and servers are symmetric. We show that, with probability 1, the price interval breaks up into three regions. At small (respectively large) prices, only finitely many bid (respectively ask) orders ever get fulfilled, while in the middle region all orders eventually clear. We derive equations which define the boundaries between these regions, and solve them explicitly in the case of iid uniform arrivals to obtain numeric values of the thresholds. We derive a heuristic for the distribution of the highest bid (respectively lowest ask), and present simulation data confirming it. Description: E-thesis pagination differs from hardbound copy kept in the Manuscripts Department, Cambridge University Library.

Modelling non-linear exposure-disease relationships in a large individual participant meta-analysis allowing for the effects of exposure measurement error

Mon, 08 Oct 2012 23:00:00 GMT

Title: Modelling non-linear exposure-disease relationships in a large individual participant meta-analysis allowing for the effects of exposure measurement error Authors: Strawbridge, Alexander Daniel Abstract: This thesis was motivated by data from the Emerging Risk Factors Collaboration (ERFC), a large individual participant data (IPD) meta-analysis of risk factors for coronary heart disease(CHD). Cardiovascular disease is the largest cause of death in almost all countries in the world, therefore it is important to be able to characterise the shape of risk factor–CHD relationships. Many of the risk factors for CHD considered by the ERFC are subject to substantial measurement error, and their relationship with CHD non-linear. We firstly consider issues associated with modelling the risk factor–disease relationship in a single study, before using meta-analysis to combine relationships across studies. It is well known that classical measurement error generally attenuates linear exposure–disease relationships, however its precise effect on non-linear relationships is less well understood. We investigate the effect of classical measurement error on the shape of exposure–disease relationships that are commonly encountered in epidemiological studies, and then consider methods for correcting for classical measurement error. We propose the application of a widely used correction method, regression calibration, to fractional polynomial models. We also consider the effects of non-classical error on the observed exposure–disease relationship, and the impact on our correction methods when we erroneously assume classical measurement error. Analyses performed using categorised continuous exposures are common in epidemiology. We show that MacMahon’s method for correcting for measurement error in analyses that use categorised continuous exposures, although simple, does not provide the correct shape for nonlinear exposure–disease relationships. We perform a simulation study to compare alternative methods for categorised continuous exposures. Meta-analysis is the statistical synthesis of results from a number of studies addressing similar research hypotheses. The use of IPD is the gold standard approach because it allows for consistent analysis of the exposure–disease relationship across studies. Methods have recently been proposed for combining non-linear relationships across studies. We discuss these methods, extend them to P-spline models, and consider alternative methods of combining relationships across studies. We apply the methods developed to the relationships of fasting blood glucose and lipoprotein(a) with CHD, using data from the ERFC.

Linear waves on higher dimensional Schwarzschild black holes and Schwarzschild de Sitter spacetimes

Mon, 02 Jul 2012 23:00:00 GMT

Title: Linear waves on higher dimensional Schwarzschild black holes and Schwarzschild de Sitter spacetimes Authors: Schlue, Volker Abstract: I study linear waves on higher dimensional Schwarzschild black holes and Schwarzschild de Sitter spacetimes. In the first part of this thesis two decay results are proven for general finite energy solutions to the linear wave equation on higher dimensional Schwarzschild black holes. I establish uniform energy decay and improved interior first order energy decay in all dimensions with rates in accordance with the 3 + 1-dimensional case. The method of proof departs from earlier work on this problem. I apply and extend the new physical space approach to decay of Dafermos and Rodnianski. An integrated local energy decay estimate for the wave equation on higher dimensional Schwarzschild black holes is proven. In the second part of this thesis the global study of solutions to the linear wave equation on expanding de Sitter and Schwarzschild de Sitter spacetimes is initiated. I show that finite energy solutions to the initial value problem are globally bounded and have a limit on the future boundary that can be viewed as a function on the standard cylinder. Both problems are related to the Cauchy problem in General Relativity.

Problems of optimal choice on posets and generalizations of acyclic colourings

Tue, 31 May 2011 23:00:00 GMT

Title: Problems of optimal choice on posets and generalizations of acyclic colourings Authors: Garrod, Bryn James Abstract: NOTE : The mathematical symbols in the abstract do not always display correctly in this text field. Please see the abstract in the thesis for the definitive abstract. ABSTRACT: This dissertation is in two parts, each of three chapters. In Part 1, I shall prove some results concerning variants of the `secretary problem'. In Part 2, I shall bound several generalizations of the acyclic chromatic number of a graph as functions of its maximum degree. I shall begin Chapter 1 by describing the classical secretary problem, in which the aim is to select the best candidate for the post of a secretary, and its solution. I shall then summarize some of its many generalizations that have been studied up to now, provide some basic theory, and briefly outline the results that I shall prove. In Chapter 2, I shall suppose that the candidates come as ‘m’ pairs of equally qualified identical twins. I shall describe an optimal strategy, a formula for its probability of success and the asymptotic behaviour of this strategy and its probability of success as m → ∞. I shall also find an optimal strategy and its probability of success for the analagous version with ‘c’-tuplets. I shall move away from known posets in Chapter 3, assuming instead that the candidates come from a poset about which the only information known is its size and number of maximal elements. I shall show that, given this information, there is an algorithm that is successful with probability at least ¹/e . For posets with ‘k ≥ 2’ maximal elements, I shall prove that if their width is also ‘k’ then this can be improved to ‘k-1√1/k’ and show that no better bound of this type is possible. In Chapter 4, I shall describe the history of acyclic colourings, in which a graph must be properly coloured with no two-coloured cycle, and state some results known about them and their variants. In particular, I shall highlight a result of Alon, McDiarmid and Reed, which bounds the acyclic chromatic number of a graph by a function of its maximum degree. My results in the next two chapters are of this form. I shall consider two natural generalizations in Chapter 5. In the first, only cycles of length at least ’l’ must receive at least three colours. In the second, every cycle must receive at least ‘c’ colours, except those of length less than ‘c’, which must be multicoloured. My results in Chapter 6 generalize the concept of a cycle; it is now subgraphs with minimum degree ‘r’ that must receive at least three colours, rather than subgraphs with minimum degree two (which contain cycles). I shall also consider a natural version of this problem for hypergraphs.

The Fukaya category, exotic forms and exotic autoequivalences

Mon, 09 Apr 2012 23:00:00 GMT

Title: The Fukaya category, exotic forms and exotic autoequivalences Authors: Harris, Richard Abstract: A symplectic manifold is a smooth manifold M together with a choice of a closed non-degenerate two-form. Recent years have seen the importance of associating an A∞-category to M, called its Fukaya category, in helping to understand symplectic properties of M and its Lagrangian submanifolds. One of the principles of this construction is that automorphisms of the symplectic manifold should induce autoequivalences of the derived Fukaya category, although precisely what autoequivalences are thus obtained has been established in very few cases. Given a Lagrangian V ≅ CPn in a symplectic manifold (M,ω), there is an associated symplectomorphism ∅v of M. In Part I, we defi ne the notion of a CPn-object in an A∞-category A, and use this to construct algebraically an A∞- functor Φv , which we prove induces an autoequivalence of the derived category DA. We conjecture that Φv corresponds to the action of ∅v and prove this in the lowest dimension n = 1. We also give examples of symplectic manifolds for which this twist can be defi ned algebraically, but corresponds to no geometric automorphism of the manifold itself: we call such autoequivalences exotic. Computations in Fukaya categories have also been useful in distinguishing certain symplectic forms on exact symplectic manifolds from the "standard" forms. In Part II, we investigate the uniqueness of so-called exotic structures on certain exact symplectic manifolds by looking at how their symplectic properties change under small nonexact deformations of the symplectic form. This allows us to distinguish between two exotic symplectic forms on T*S3∪2-handle, even though the standard symplectic invariants such as their Fukaya category and their symplectic cohomology vanish. We also exhibit, for any n, an exact symplectic manifold with n distinct, exotic symplectic structures, which again cannot be distinguished by symplectic cohomology or by the Fukaya category.

Hyper and structural Markov laws for graphical models

Tue, 06 Mar 2012 00:00:00 GMT

Title: Hyper and structural Markov laws for graphical models Authors: Byrne, Simon Abstract: My thesis focuses on the parameterisation and estimation of graphical models, based on the concept of hyper and meta Markov properties. These state that the parameters should exhibit conditional independencies, similar to those on the sample space. When these properties are satisfied, parameter estimation may be performed locally, i.e. the estimators for certain subsets of the graph are determined entirely by the data corresponding to the subset. Firstly, I discuss the applications of these properties to the analysis of case-control studies. It has long been established that the maximum likelihood estimates for the odds-ratio may be found by logistic regression, in other words, the "incorrect" prospective model is equivalent to the correct retrospective model. I use a generalisation of the hyper Markov properties to identify necessary and sufficient conditions for the corresponding result in a Bayesian analysis, that is, the posterior distribution for the odds-ratio is the same under both the prospective and retrospective likelihoods. These conditions can be used to derive a parametric family of prior laws that may be used for such an analysis. The second part focuses on the problem of inferring the structure of the underlying graph. I propose an extension of the meta and hyper Markov properties, which I term structural Markov properties, for both undirected decomposable graphs and directed acyclic graphs. Roughly speaking, it requires that the structure of distinct components of the graph are conditionally independent given the existence of a separating component. This allows the analysis and comparison of multiple graphical structures, while being able to take advantage of the common conditional independence constraints. Moreover, I show that these properties characterise exponential families, which form conjugate priors under sampling from compatible Markov distributions.

Statistical issues in Mendelian randomization: use of genetic instrumental variables for assessing causal associations

Tue, 06 Mar 2012 00:00:00 GMT

Title: Statistical issues in Mendelian randomization: use of genetic instrumental variables for assessing causal associations Authors: Burgess, Stephen Abstract: Mendelian randomization is an epidemiological method for using genetic variation to estimate the causal effect of the change in a modifiable phenotype on an outcome from observational data. A genetic variant satisfying the assumptions of an instrumental variable for the phenotype of interest can be used to divide a population into subgroups which differ systematically only in the phenotype. This gives a causal estimate which is asymptotically free of bias from confounding and reverse causation. However, the variance of the causal estimate is large compared to traditional regression methods, requiring large amounts of data and necessitating methods for efficient data synthesis. Additionally, if the association between the genetic variant and the phenotype is not strong, then the causal estimates will be biased due to the “weak instrument” in finite samples in the direction of the observational association. This bias may convince a researcher that an observed association is causal. If the causal parameter estimated is an odds ratio, then the parameter of association will differ depending on whether viewed as a population-averaged causal effect or a personal causal effect conditional on covariates. We introduce a Bayesian framework for instrumental variable analysis, which is less susceptible to weak instrument bias than traditional two-stage methods, has correct coverage with weak instruments, and is able to efficiently combine gene–phenotype–outcome data from multiple heterogeneous sources. Methods for imputing missing genetic data are developed, allowing multiple genetic variants to be used without reduction in sample size. We focus on the question of a binary outcome, illustrating how the collapsing of the odds ratio over heterogeneous strata in the population means that the two-stage and the Bayesian methods estimate a population-averaged marginal causal effect similar to that estimated by a randomized trial, but which typically differs from the conditional effect estimated by standard regression methods. We show how these methods can be adjusted to give an estimate closer to the conditional effect. We apply the methods and techniques discussed to data on the causal effect of C-reactive protein on fibrinogen and coronary heart disease, concluding with an overall estimate of causal association based on the totality of available data from 42 studies.

Unipotent elements in algebraic groups

Tue, 10 Jan 2012 00:00:00 GMT

Title: Unipotent elements in algebraic groups Authors: Clarke, Matthew Charles Abstract: This thesis is concerned with three distinct, but closely related, research topics focusing on the unipotent elements of a connected reductive algebraic group G, over an algebraically closed field k, and nilpotent elements in the Lie algebra g = LieG. The first topic is a determination of canonical forms for unipotent classes and nilpotent orbits of G. Using an original approach, we begin by obtaining a new canonical form for nilpotent matrices, up to similarity, which is symmetric with respect to the non-main diagonal (i.e. it is fixed by the map f : (xi;j) -> (xn+1-j;n+1-i)), with entries in {0,1}. We then show how to modify this form slightly in order to satisfy a non-degenerate symmetric or skew-symmetric bilinear form, assuming that the orbit does not vanish in the presence of such a form. Replacing G by any simple classical algebraic group, we thus obtain a unified approach to computing representatives for nilpotent orbits for all classical groups G. By applying Springer morphisms, this also yields representatives for the corresponding unipotent classes in G. As a corollary, we obtain a complete set of generic canonical representatives for the unipotent classes of the finite general unitary groups GUn(Fq) for all prime powers q. Our second topic is concerned with unipotent pieces, defined by G. Lusztig in [Unipotent elements in small characteristic, Transform. Groups 10 (2005), 449-487]. We give a case-free proof of the conjectures of Lusztig from that paper. This presents a uniform picture of the unipotent elements of G, which can be viewed as an extension of the Dynkin-Kostant theory, but is valid without restriction on p. We also obtain analogous results for the adjoint action of G on its Lie algebra g and the coadjoint action of G on g*. We also obtain several general results about the Hesselink stratification and Fq-rational structures on G-modules. Our third topic is concerned with generalised Gelfand-Graev representations of finite groups of Lie type. Let u be a unipotent element in such a group GF and let Γu be the associated generalised Gelfand-Graev representation of GF . Under the assumption that G has a connected centre, we show that the dimension of the endomorphism algebra of Γu is a polynomial in q (the order of the associated finite field), with degree given by dimCG(u). When the centre of G is disconnected, it is impossible, in general, to parametrise the (isomorphism classes of) generalised Gelfand-Graev representations independently of q, unless one adopts a convention of considering separately various congruence classes of q. Subject to such a convention, we extend our result. We also present computational data related to the main theoretical results. In particular, tables of our canonical forms are given in the appendices, as well as tables of dimension polynomials for endomorphism algebras of generalised Gelfand-Graev representations, together with the relevant GAP source code.

Witt groups of complex varieties

Mon, 11 Jul 2011 23:00:00 GMT

Title: Witt groups of complex varieties Authors: Zibrowius, Marcus Abstract: The thesis Witt Groups of Complex Varieties studies and compares two related cohomology theories that arise in the areas of algebraic geometry and topology: the algebraic theory of Witt groups, and real topological K-theory. Specifically, we introduce comparison maps from the Grothendieck-Witt and Witt groups of a smooth complex variety to the KO-groups of the underlying topological space and analyse their behaviour. We focus on two particularly favourable situations. Firstly, we explicitly compute the Witt groups of smooth complex curves and surfaces. Using the theory of Stiefel-Whitney classes, we obtain a satisfactory description of the comparison maps in these low-dimensional cases. Secondly, we show that the comparison maps are isomorphisms for smooth cellular varieties. This result applies in particular to projective homogeneous spaces. By extending known computations in topology, we obtain an additive description of the Witt groups of all projective homogeneous varieties that fall within the class of hermitian symmetric spaces.

Cliques in graphs

Mon, 11 Oct 2010 23:00:00 GMT

Title: Cliques in graphs Authors: Lo, Allan Abstract: The main focus of this thesis is to evaluate $k_r(n,\delta)$, the minimal number of $r$-cliques in graphs with $n$ vertices and minimum degree~$\delta$. A fundamental result in Graph Theory states that a triangle-free graph of order $n$ has at most $n^2/4$ edges. Hence, a triangle-free graph has minimum degree at most $n/2$, so if $k_3(n,\delta) =0$ then $\delta \le n/2$. For $n/2 \leq \delta \leq 4n/5$, I have evaluated $k_r(n,\delta)$ and determined the structures of the extremal graphs. For $\delta \ge 4n/5$, I give a conjecture on $k_r(n,\delta)$, as well as the structures of these extremal graphs. Moreover, I have proved various partial results that support this conjecture. Let $k_r^{reg}(n, \delta)$ be the analogous version of $k_r(n,\delta)$ for regular graphs. Notice that there exist $n$ and $\delta$ such that $k_r(n, \delta) =0$ but $k_r^{reg}(n, \delta) >0$. For example, a theorem of Andr{\'a}sfai, Erd{\H{o}}s and S{\'o}s states that any triangle-free graph of order $n$ with minimum degree greater than $2n/5$ must be bipartite. Hence $k_3(n, \lfloor n/2 \rfloor) =0$ but $k_3^{reg}(n, \lfloor n/2 \rfloor) >0$ for $n$ odd. I have evaluated the exact value $k_3^{reg}(n, \delta)$ for $\delta$ between $2n/5+12 \sqrt{n}/5$ and $n/2$ and determined the structure of these extremal graphs. At the end of the thesis, I investigate a question in Ramsey Theory. The Ramsey number $R_k(G)$ of a graph $G$ is the minimum number $N$, such that any edge colouring of $K_N$ with $k$ colours contains a monochromatic copy of $G$. The constrained Ramsey number $f(G,T)$ of two graphs $G$ and $T$ is the minimum number $N$ such that any edge colouring of $K_N$ with any number of colours contains a monochromatic copy of $G$ or a rainbow copy of $T$. It turns out that these two quantities are closely related when $T$ is a matching. Namely, for almost all graphs $G$, $f(G,tK_2) =R_{t-1}(G)$ for $t \geq 2$.

Maximum likelihood estimation of a multivariate log-concave density

Tue, 12 Jan 2010 00:00:00 GMT

Title: Maximum likelihood estimation of a multivariate log-concave density Authors: Cule, Madeleine Abstract: Density estimation is a fundamental statistical problem. Many methods are either sensitive to model misspecification (parametric models) or difficult to calibrate, especially for multivariate data (nonparametric smoothing methods). We propose an alternative approach using maximum likelihood under a qualitative assumption on the shape of the density, specifically log-concavity. The class of log-concave densities includes many common parametric families and has desirable properties. For univariate data, these estimators are relatively well understood, and are gaining in popularity in theory and practice. We discuss extensions for multivariate data, which require different techniques. After establishing existence and uniqueness of the log-concave maximum likelihood estimator for multivariate data, we see that a reformulation allows us to compute it using standard convex optimization techniques. Unlike kernel density estimation, or other nonparametric smoothing methods, this is a fully automatic procedure, and no additional tuning parameters are required. Since the assumption of log-concavity is non-trivial, we introduce a method for assessing the suitability of this shape constraint and apply it to several simulated datasets and one real dataset. Density estimation is often one stage in a more complicated statistical procedure. With this in mind, we show how the estimator may be used for plug-in estimation of statistical functionals. A second important extension is the use of log-concave components in mixture models. We illustrate how we may use an EM-style algorithm to fit mixture models where the number of components is known. Applications to visualization and classification are presented. In the latter case, improvement over a Gaussian mixture model is demonstrated. Performance for density estimation is evaluated in two ways. Firstly, we consider Hellinger convergence (the usual metric of theoretical convergence results for nonparametric maximum likelihood estimators). We prove consistency with respect to this metric and heuristically discuss rates of convergence and model misspecification, supported by empirical investigation. Secondly, we use the mean integrated squared error to demonstrate favourable performance compared with kernel density estimates using a variety of bandwidth selectors, including sophisticated adaptive methods. Throughout, we emphasise the development of stable numerical procedures able to handle the additional complexity of multivariate data.

Arithmetic structure in sets of integers

Mon, 19 May 2008 23:00:00 GMT

Title: Arithmetic structure in sets of integers Authors: Wolf, Julia Abstract: This dissertation deals with four problems concerning arithmetic structures in dense sets of integers. In Chapter 1 we give an exposition of the state-of-the-art technique due to Pintz, Steiger and Szemer edi which yields the best known upper bound on the density of sets whose di erence set is square-free. Inspired by the well-known fact that Fourier analysis is not su cient to detect progressions of length 4 or more, we determine in Chapter 2 a necessary and sufficient condition on a system of linear equations which guarantees the correct number of solutions in any uniform subset of Fnp. This joint work with Tim Gowers constitutes the core of this thesis and relies heavily on recent progress in so-called "quadratic Fourier analysis" pioneered by Gowers, Green and Tao. In particular, we use a structure theorem for bounded functions which provides a decomposition into a quadratically structured and a quadratically uniform part. We also present an alternative decomposition leading to improved bounds for the main result, and discuss the connections with recent results in ergodic theory. Chapter 3 deals with improved upper and lower bounds on the minimum number of monochromatic 4-term progressions in any two-colouring of ZN. Finally, in Chapter 4 we investigate the structure of the set of popular di erences of a subset of ZN. More precisely, we establish that, given a subset of size linear in N, the set of its popular differences does not always contain the complete difference set of another large set.

Topics in arithmetic combinatorics

Mon, 22 Oct 2007 23:00:00 GMT

Title: Topics in arithmetic combinatorics Authors: Sanders, Tom Abstract: This thesis is chiefly concerned with a classical conjecture of Littlewood's regarding the L^1-norm of the Fourier transform, and the closely related idempotent theorem. The vast majority of the results regarding these problems are, in some sense, qualitative or at the very least infinitary and it has become increasingly apparent that a quantitative state of affairs is desirable. Broadly speaking, the first part of the thesis develops three new tools for tackling the problems above: We prove a new structural theorem for the spectrum of functions in A(G); we extend the notion of local Fourier analysis, pioneered by Bourgain, to a much more general structure, and localize Chang's classic structure theorem as well as our own spectral structure theorem; and we refine some aspects of Freiman's celebrated theorem regarding the structure of sets with small doubling. These tools lead to improvements in a number of existing additive results which we indicate, but for us the main purpose is in application to the analytic problems mentioned above. The second part of the thesis discusses a natural version of Littlewood's problem for finite abelian groups. Here the situation varies wildly with the underlying group and we pay special attention first to the finite field case (where we use Chang's Theorem) and then to the case of residues modulo a prime where we require our new local structure theorem for A(G). We complete the consideration of Littlewood's problem for finite abelian groups by using the local version of Chang's Theorem we have developed. Finally we deploy the Freiman tools along with the extended Fourier analytic techniques to yield a fully quantitative version of the idempotent theorem. Description: E-thesis pagination differs from approved hard bound copy, Cambridge University Library classmark: PhD.30726

Iwasawa theory for modular forms at supersingular primes

Mon, 05 Jul 2010 23:00:00 GMT

Title: Iwasawa theory for modular forms at supersingular primes Authors: Lei, Antonio Abstract: Let f=\sum a_nq^n be a normalised eigen-newform of weight k\ge2 and p an odd prime which does not divide the level of f. We study a reformulation of Kato's main conjecture for f over the Zp-cyclotomic extension of Q. In particular, we generalise Kobayashi's main conjecture on p-supersingular elliptic curves over Q with a_p=0, which asserts that Pollack's p-adic L-functions generate the characteristic ideals of some \pm-Selmer groups which are cotorsion over the Iwasawa algebra \Lambda=Zp[[Zp]]. We begin by studying the p-adic Hodge theory for the p-adic representation associated to f in the case when a_p=0. It allows us to give analogous definitions of Kobayashi's \pm-Coleman maps and \pm-Selmer groups. The Coleman maps are used to show that the Pontryagin duals of these new Selmer groups are torsion over \Lambda as in the elliptic curve case. As a consequence, we formulate a main conjecture stating that Pollack's p-adic L-functions generate their characteristic ideals. Similar to Kobayashi's works, we prove one inclusion of the main conjecture using an Euler system constructed by Kato. We then prove the other inclusion of the main conjecture for CM modular forms, generalising works of Pollack and Rubin on CM elliptic curves. As a key step of the proof, we generalise the reciprocity law of Coates-Wiles and Rubin. Next, we study Wach modules associated to positive crystalline p-adic representations in general and generalise the construction of the Coleman maps. By applying this to modular forms with much more general a_p, we define two Coleman maps and decompose the classical p-adic L functions of f into linear combinations of two power series of bounded coefficients generalising works of Pollack (in the case a_p=0) and Sprung (when f corresponds to an elliptic curve over Q with a_p\ne0). Once again, this leads to a reformulation of Kato's main conjecture involving cotorsion Selmer groups and p-adic L-functions of bounded coefficients. One inclusion of this new main conjecture is proved in the same way as the a_p=0 case. Finally, we explain how the \pm-Coleman maps can be extended to Lubin-Tate extensions of height 1 in place of the Zp-cyclotomic extension. This generalises works of Iovita and Pollack for elliptic curves over Q.

Non-commutative Iwasawa theory of elliptic curves at primes of multiplicative reduction

Mon, 05 Jul 2010 23:00:00 GMT

Title: Non-commutative Iwasawa theory of elliptic curves at primes of multiplicative reduction Authors: Lee, Chern-Yang Abstract: Let E be an elliptic curve defined over the rationals Q, and p be a prime at least 5 where E has multiplicative reduction. This thesis studies the Iwasawa theory of E over certain false Tate curve extensions F[infinity], with Galois group G = Gal(F[infinity]/Q). I show how the p[infinity]-Selmer group of E over F[infinity] controls the p[infinity]-Selmer rank growth within the false Tate curve extension, and how it is connected to the root numbers of E twisted by absolutely irreducible orthogonal Artin representations of G, and investigate the parity conjecture for twisted modules.

Models of genus one curves

Tue, 16 Mar 2010 00:00:00 GMT

Title: Models of genus one curves Authors: Sadek, Mohammad Abstract: In this thesis we give insight into the minimisation problem of genus one curves defined by equations other than Weierstrass equations. We are interested in genus one curves given as double covers of P1, plane cubics, or complete intersections of two quadrics in P3. By minimising such a curve we mean making the invariants associated to its defining equations as small as possible using a suitable change of coordinates. We study the non-uniqueness of minimisations of the genus one curves described above. To achieve this goal we investigate models of genus one curves over Henselian discrete valuation rings. We give geometric criteria which relate these models to the minimal proper regular models of the Jacobian elliptic curves of the genus one curves above. We perform explicit computations on the special fibers of minimal proper regular models of elliptic curves. Then we use these computations to count the number of minimisations of a genus one curve defined over a Henselian discrete valuation field. This number depends only on the Kodaira symbol of the Jacobian and on an auxiliary rational point. Finally, we consider the minimisation problem of a genus one curve defined over Q.

On eigenvectors for semisimple elements in actions of algebraic groups

Tue, 09 Feb 2010 00:00:00 GMT

Title: On eigenvectors for semisimple elements in actions of algebraic groups Authors: Kenneally, Darren John Abstract: Let $G$ be a simple simply connected algebraic group defined over an algebraically closed field $K$ and $V$ an irreducible module defined over $K$ on which $G$ acts. Let $E$ denote the set of vectors in $V$ which are eigenvectors for some non-central semisimple element of $G$ and some eigenvalue in $K^∗$. We prove, with a short list of possible exceptions, that the dimension of $\overline{E}$ is strictly less than the dimension of $V$ provided $\dim V > \dim G + 2$ and that there is equality otherwise. In particular, by considering only the eigenvalue $1$, it follows that the closure of the union of fixed point spaces of non-central semisimple elements has dimension strictly less than the dimension of $V$ provided $\dim V > \dim G + 2$, with a short list of possible exceptions. In the majority of cases we consider modules for which $\dim V > \dim G + 2$ where we perform an analysis of weights. In many of these cases we prove that, for any non-central semisimple element and any eigenvalue, the codimension of the eigenspace exceeds $\dim G$. In more difficult cases, when $\dim V$ is only slightly larger than $\dim G + 2$, we subdivide the analysis according to the type of the centraliser of the semisimple element. Here we prove for each type a slightly weaker inequality which still suffices to establish the main result. Finally, for the relatively few modules satisfying $\dim V \leq \dim G + 2$, an immediate observation yields the result for $\dim V < \dim B$ where $B$ is a Borel subgroup of $G$, while in other cases we argue directly.

Graphical representations of Ising and Potts models: stochastic geometry of the quantum Ising model and the space-time Potts model

Tue, 09 Feb 2010 00:00:00 GMT

Title: Graphical representations of Ising and Potts models: stochastic geometry of the quantum Ising model and the space-time Potts model Authors: Björnberg, Jakob Erik Abstract: Statistical physics seeks to explain macroscopic properties of matter in terms of microscopic interactions. Of particular interest is the phenomenon of phase transition: the sudden changes in macroscopic properties as external conditions are varied. Two models in particular are of great interest to mathematicians, namely the Ising model of a magnet and the percolation model of a porous solid. These models in turn are part of the unifying framework of the random-cluster representation, a model for random graphs which was first studied by Fortuin and Kasteleyn in the 1970’s. The random-cluster representation has proved extremely useful in proving important facts about the Ising model and similar models. In this work we study the corresponding graphical framework for two related models. The first model is the transverse field quantum Ising model, an extension of the original Ising model which was introduced by Lieb, Schultz and Mattis in the 1960’s. The second model is the space–time percolation process, which is closely related to the contact model for the spread of disease. In Chapter 2 we define the appropriate ‘space–time’ random-cluster model and explore a range of useful probabilistic techniques for studying it. The space–time Potts model emerges as a natural generalization of the quantum Ising model. The basic properties of the phase transitions in these models are treated in this chapter, such as the fact that there is at most one unbounded fk-cluster, and the resulting lower bound on the critical value in Z. In Chapter 3 we develop an alternative graphical representation of the quantum Ising model, called the random-parity representation. This representation is based on the random-current representation of the classical Ising model, and allows us to study in much greater detail the phase transition and critical behaviour. A major aim of this chapter is to prove sharpness of the phase transition in the quantum Ising model—a central issue in the theory—and to establish bounds on some critical exponents. We address these issues by using the random-parity representation to establish certain differential inequalities, integration of which give the results. In Chapter 4 we explore some consequences and possible extensions of the results established in Chapters 2 and 3. For example, we determine the critical point for the quantum Ising model in Z and in ‘star-like’ geometries.

Three viewpoints on semi-abelian homology

Tue, 17 Nov 2009 00:00:00 GMT

Title: Three viewpoints on semi-abelian homology Authors: Goedecke, Julia Abstract: The main theme of the thesis is to present and compare three different viewpoints on semi-abelian homology, resulting in three ways of defining and calculating homology objects. Any two of these three homology theories coincide whenever they are both defined, but having these different approaches available makes it possible to choose the most appropriate one in any given situation, and their respective strengths complement each other to give powerful homological tools. The oldest viewpoint, which is borrowed from the abelian context where it was introduced by Barr and Beck, is comonadic homology, generating projective simplicial resolutions in a functorial way. This concept only works in monadic semi-abelian categories, such as semi-abelian varieties, including the categories of groups and Lie algebras. Comonadic homology can be viewed not only as a functor in the first entry, giving homology of objects for a particular choice of coefficients, but also as a functor in the second variable, varying the coefficients themselves. As such it has certain universality properties which single it out amongst theories of a similar kind. This is well-known in the setting of abelian categories, but here we extend this result to our semi-abelian context. Fixing the choice of coefficients again, the question naturally arises of how the homology theory depends on the chosen comonad. Again it is well-known in the abelian case that the theory only depends on the projective class which the comonad generates. We extend this to the semi-abelian setting by proving a comparison theorem for simplicial resolutions. This leads to the result that any two projective simplicial resolutions, the definition of which requires slightly more care in the semi-abelian setting, give rise to the same homology. Thus again the homology theory only depends on the projective class. The second viewpoint uses Hopf formulae to define homology, and works in a non-monadic setting; it only requires a semi-abelian category with enough projectives. Even this slightly weaker setting leads to strong results such as a long exact homology sequence, the Everaert sequence, which is a generalised and extended version of the Stallings-Stammbach sequence known for groups. Hopf formulae use projective presentations of objects, and this is closer to the abelian philosophy of using any projective resolution, rather than a special functorial one generated by a comonad. To define higher Hopf formulae for the higher homology objects the use of categorical Galois theory is crucial. This theory allows a choice of Birkhoff subcategory to generate a class of central extensions, which play a big role not only in the definition via Hopf formulae but also in our third viewpoint. This final and new viewpoint we consider is homology via satellites or pointwise Kan extensions. This makes the universal properties of the homology objects apparent, giving a useful new tool in dealing with statements about homology. The driving motivation behind this point of view is the Everaert sequence mentioned above. Janelidze's theory of generalised satellites enables us to use the universal properties of the Everaert sequence to interpret homology as a pointwise Kan extension, or limit. In the first instance, this allows us to calculate homology step by step, and it removes the need for projective objects from the definition. Furthermore, we show that homology is the limit of the diagram consisting of the kernels of all central extensions of a given object, which forges a strong connection between homology and cohomology. When enough projectives are available, we can interpret homology as calculating fixed points of endomorphisms of a given projective presentation.