Pure Mathematics http://hdl.handle.net/10012/9932 2019-06-25T16:19:26Z Approximation Constants for Closed Subschemes of Projective Varieties http://hdl.handle.net/10012/14764 Approximation Constants for Closed Subschemes of Projective Varieties Rollick, Nickolas Diophantine approximation is a branch of number theory with a long history, going back at least to the work of Dirichlet and Liouville in the 1840s. The innocent-looking question of how well an arbitrary real algebraic number can be approximated by rational numbers (relative to the size of the denominator of the approximating rational number) took more than 100 years to resolve, culminating in the definitive Fields Medal-winning work of Klaus Roth in 1955. Much more recently, David McKinnon and Mike Roth have re-phrased and generalized this Diophantine approximation question to apply in the setting of approximating algebraic points on projective varieties defined over number fields. To do this, they defined an "approximation constant", depending on the point one wishes to approximate and a given line bundle. This constant measures the tradeoff between the closeness of the approximation and the arithmetic complexity of the point used to make the approximation, as measured by a height function associated to the line bundle. In particular, McKinnon and Roth succeeded in proving lower bounds on the approximation constant in terms of the "Seshadri constant" associated to the given point and line bundle, measuring local positivity of the line bundle around the point. Appropriately interpreted, these results generalize the classical work of Liouville and Roth, and the corresponding McKinnon-Roth theorems are therefore labelled "Liouville-type" and "Roth-type" results. Recent work of Grieve and of Ru-Wang have taken the Roth-type theorems even further; in contrast, we explore results of Liouville-type, which are more elementary in nature. In Chapter 2, we lay the groundwork necessary to define the approximation constant at a point, before generalizing the McKinnon-Roth definition to approximations of arbitrary closed subschemes. We also introduce the notion of an essential approximation constant, which ignores unusually good approximations along proper Zariski-closed subsets. After verifying that our new approximation constant truly does generalize the constant of McKinnon-Roth, Chapter 3 establishes a fundamental lower bound on the approximation constants of closed subschemes of projective space, depending only on the equations cutting out the subscheme. In Chapter 4, we provide a series of explicit computations of approximation constants, both for subschemes satisfying suitable geometric conditions, and for curves of low degree in projective 3-space. We will encounter difficulties computing the approximation constant exactly for general cubic curves, and we spend some time showing why some of the more evident approaches do not succeed. To conclude the chapter, we take up the question of large gaps between the ordinary and essential approximation constants, by considering approximations to a certain rational point on a diagonal quartic surface. Finally, in Chapter 5, we generalize the Liouville-type results of McKinnon-Roth. 2019-06-19T00:00:00Z Degrees of Categoricity and the Isomorphism Problem http://hdl.handle.net/10012/14753 Degrees of Categoricity and the Isomorphism Problem Mahmoud, Mohammad In this thesis, we study notions of complexity related to computable structures. We first study degrees of categoricity for computable tree structures. We show that, for any computable ordinal $\alpha$, there exists a computable tree of rank $\alpha+1$ with strong degree of categoricity ${\bf 0}^{(2\alpha)}$ if $\alpha$ is finite, and with strong degree of categoricity ${\bf 0}^{(2\alpha+1)}$ if $\alpha$ is infinite. For a computable limit ordinal $\alpha$, we show that there is a computable tree of rank $\alpha$ with strong degree of categoricity ${\bf 0}^{(\alpha)}$ (which equals ${\bf 0}^{(2\alpha)}$). In general, it is not the case that every Turing degree is the degree of categoricity of some structure. However, it is known that every degree that is of a computably enumerable (c.e.) set\ in and above $\mathbf{0}^{(\alpha)}$, for $\alpha$ a successor ordinal, is a degree of categoricity. In this thesis, we include joint work with Csima, Deveau and Harrison-Trainor which shows that every degree c.e.\ in and above $\mathbf{0}^{(\alpha)}$, for $\alpha$ a limit ordinal, is a degree of categoricity. We also show that every degree c.e.\ in and above $\mathbf{0}^{(\omega)}$ is the degree of categoricity of a prime model, making progress towards a question of Bazhenov and Marchuk. After that, we study the isomorphism problem for tree structures. It follows from our proofs regarding the degrees of categoricity for these structures that, for every computable ordinal $\alpha>0$, the isomorphism problem for trees of rank $\alpha$ is $\Pi_{2\alpha}$-complete. We also discuss the isomorphism problem for pregeometries in which dependent elements are dense and the closure operator is relatively intrinsically computably enumerable. We show that, if $K$ is a class of such pregeometries, then the isomorphism problem for the class $K$ is $\Pi_3$-hard. Finally, we study the Turing ordinal. We observed that the definition of the Turing ordinal has two parts each of which alone can define a specific ordinal which we call the upper and lower Turing ordinals. The Turing ordinal exists if and only if these two ordinals exist and are equal. We give examples of classes of computable structures such that the upper Turing ordinal is $\beta$ and the lower Turing ordinal is $\alpha$ for all computable ordinals $\alpha<\beta$. 2019-06-12T00:00:00Z Sidon and Kronecker-like sets in compact abelian groups http://hdl.handle.net/10012/14746 Sidon and Kronecker-like sets in compact abelian groups Yang, Xu Let $G$ be a compact abelian group and $\Ga$ be its discrete dual group. In this thesis we study various types of interpolation sets. A subset $E \subset \Ga$ is Sidon if every bounded function on $E$ can be interpolated by the Fourier transform of a finite complex measure. Sidon sets have been extensively studied, and one significant breakthrough, that Sidonicity is equivalent to proportional quasi-independence, was proved by Bourgain and Pisier during the early 80s. In this thesis we will give a detailed demonstration of Pisier's approach. We also seek for possible extensions of Pisier's theorems. Based on Pisier's techniques, we will show Sidonicity is equivalent to proportional independence of higher degrees and minimal constants. A subset $E \subset \Ga$ is $\e$-Kronecker if every function on $E$ with range in the unit circle can be interpolated by a continuous character on $\Ga$ with an error of $\e$. We will prove some interesting properties of $\e$-Kronecker sets and give an estimation of the Sidon constant of such sets. Generalizations of Kronecker sets include binary Kronecker sets and $N$-pseudo-Rademacher sets. We compute the binary Kronecker constants of some interesting examples. For $N$-pseudo-Rademacher sets, we give a characterization of such sets, describe their structures and prove the existence of large $N$-pseudo-Rademacher sets. 2019-06-07T00:00:00Z The Logarithmic Derivative and Model-Theoretic Analysability in Differentially Closed Fields http://hdl.handle.net/10012/14379 The Logarithmic Derivative and Model-Theoretic Analysability in Differentially Closed Fields Jin, Ruizhang This thesis deals with internal and analysable types, mainly in the context of the stable theory of differentially closed fields. Two main problems are dealt with: the construction of types analysable in the constants with specific properties, and a criterion for a given analysable type to be actually internal to the constants. For analysable types, the notion of canonical analyses is introduced. A type has a canonical analysis if all its analyses of shortest length are interalgebraic. Given a finite sequence of ranks, it is constructed, in the theory of differentially closed field, a type analysable in the constants such that it admits a canonical analysis and each step of the analysis is of the given rank. The construction of such a type starts from the well-known example of δ(logδx)=0, whose generic type is analysable in the constants in 2 steps but is not internal to the constants. Along the way, techniques for comparing analyses in stable theories are developed, including in particular the notions of analyses by reductions and by coreductions. The property of the logδ function is further studied when the following question is raised: given a type internal to the constants, is its preimage under logδ, which is 2-step analysable in the constants, ever internal to the constants? The question is answered positively, and a criterion for when the preimage is indeed internal is proposed. Partial results are proven for this conjectured criterion, namely the cases where the group of automorphisms (the binding group) of the given internal type is additive, multiplicative, or trivial. In particular, the conjecture is resolved for generic types of equations of the form δx=f(x) where f is a rational function over the constants. It is discovered that the related problem where logδ is replaced by δ is significantly different, and the analogue of the conjecture fails in this case. Also included in this thesis are two examples asked for in the literature: internality of a particular twisted D-group, and a 2-step analysable set with independent fibres. 2019-01-22T00:00:00Z