This thesis is a model-theoretic study of exponential differential equations in the context of differential algebra. I define the theory of a set of differential equations and give an axiomatization for the theory of the exponential differential equations of split semiabelian varieties. In particular, this includes the theory of the equations satisfied by the usual complex exponential function and the Weierstrass p-functions. The theory consists of a description of the algebraic structure on the solution sets together with necessary and sufficient conditions for a system of equations to have solutions. These conditions are stated in terms of a dimension theory; their necessity generalizes Ax’s differential field version of Schanuel’s conjecture and their sufficiency generalizes recent work of Crampin. They are shown to apply to the solving of systems of equations in holomorphic functions away from singularities, as well as in the abstract setting. The theory can also be obtained by means of a Hrushovski-style amalgamation construction, and I give a category-theoretic account of the method. Restricting to the usual exponential differential equation, I show that a “blurring” of Zilber’s pseudo-exponentiation satisfies the same theory. I conjecture that this theory also holds for a suitable blurring of the complex exponential maps and partially resolve the question, proving the necessity but not the sufficiency of the aforementioned conditions. As an algebraic application, I prove a weak form of Zilber’s conjecture on intersections with subgroups (known as CIT) for semiabelian varieties. This in turn is used to show that the necessary and sufficient conditions are expressible in the appropriate first order language.

Senior Thesis at CMC is often described institutionally as the capstone of one’s undergraduate education. As such, I wanted my own to accurately capture and reflect how I’ve grown as a student and mathematician these past four years. What follows is my attempt to distill lessons I learned in mathematics outside the curriculum, written for incoming undergraduates and anyone with just a little bit of mathematical curiosity. In it, I attempt to dispel some common preconceptions about mathematics, namely that it’s uninteresting, formulaic, acultural, or completely objective, in favor of a dynamic historical and cultural perspective, with particular attention paid to the early twentieth century search to secure the foundations of mathematics and a detailed look at contemporary Hungarian mathematics. After doing so, I conclude that the scope of mathematics is not what one might expect but that it’s still absolutely worth doing and appreciating.

This thesis examines possible philosophies to account for the practice of mathematics, exploring the metaphysical, ontological, and epistemological outcomes of each possible theory. Through a study of the two most probable ideas, mathematical platonism and fictionalism, I focus on the compelling argument for platonism given by an appeal to the sciences. The Indispensability Argument establishes the power of explanation seen in the relationship between mathematics and empirical science. Cases of this explanatory power illustrate how we might have reason to believe in the existence of mathematical entities present within our best scientific theories. The second half of this discussion surveys Newtonian Cosmology and other inconsistent theories as they pose issues that have received insignificant attention within the philosophy of mathematics. The application of these inconsistent theories raises questions about the effectiveness of mathematics to model physical systems.

A suitable subcategory of affine Azumaya algebras is defined and a functor from this category to the category of Zariski structures is constructed. The rudiments of a theory of presheaves of topological structures is developed and applied to construct examples of structures at a generic parameter. The category of equivariant algebras is defined and a first-order theory is associated to each object. For those theories satisfying a certain technical condition, uncountable categoricity and quantifier elimination results are established. Models are shown to be Zariski structures and a functor from the category of equivariant algebras to Zariski structures is constructed. The two functors obtained in the thesis are shown to agree on a nontrivial class of algebras.

Considerable study has been made concerning the properties of the iterations of a homeomorphism on a metric space. Much of this material is scattered throughout the literature and understood solely by a specialist. The main object of this paper is to put into readable form proofs of theorems found in G.T. Whyburn's "Analytic Topology" pertaining to this topic in topology. Properties of the decomposition space of point-orbits induced by the iterations of a homeomorphism will compose a major part of the study. Some theorems will be established through series of lemmas required to fill in much of the detail lacking in proofs found the book. Although an elementary knowledge of topology is assumed throughout the paper, references are given for basic definitions and theorems used in developing some of the proofs. The following symbols and notation will be used throughout the paper. X will denote a metric space with metric p, S a topological space, I the set of positive integers, A, B, C... sets of points or elements. Small letters, such as a, b, c, x, y, z... will designate elements or points of sets. U and V will denote open sets Sr(x) a spherical neighborhood of x with radius r. A' denotes the set of limit points of A. A- the set of closure points of A/ U, N, C will denote union, intersection, and set inclusion respectively. The symbol E will mean "is an element of". 0 denotes the empty set. S - A is the set of points in S which are not in A.

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This thesis explores the geometric principles underlying many of the known Trichotomy Theorems. The main aims are to unify the field construction in non-linear o-minimal structures and generalizations of Zariski Geometries as well as to pave the road for completely new results in this direction. In the first part of this thesis we introduce a new axiomatic framework in which all the relevant structures can be studied uniformly and show that these axioms are preserved under elementary extensions. A particular focus is placed on the study of a smoothness condition which generalizes the presmoothness condition for Zariski Geometries. We also modify Zilber's notion of universal specializations to obtain a suitable notion of infinitesimals. In addition, families of curves and the combinatorial geometry of one-dimensional structures are studied to prove a weak trichotomy theorem based on very weak one-basedness. It is then shown that under suitable additional conditions groups and group actions can be constructed in canonical ways. This construction is based on a notion of ``geometric calculus'' and can be seen in close analogy with ordinary differentiation. If all conditions are met, a definable distributive action of one one-dimensional type-definable group on another are obtained. The main result of this thesis is that both o-minimal structures and generalizations of Zariski Geometries fit into this geometric framework and that the latter always satisfy the conditions required in the group constructions. We also exhibit known methods that allow us to extract fields from this. In addition to unifying the treatment of o-minimal structures and Zariski Geometries, this also gives a direct proof of the Trichotomy Theorem for "type-definable" Zariski Geometries as used, for example, in Hrushovski's proof of the relative Mordell-Lang conjecture.

This thesis investigates the connections between henselian valuations and absolute Galois groups. There are fundamental links between these: On one hand, the absolute Galois group of a field often encodes information about (henselian) valuations on that field. On the other, in many cases a henselian valuation imposes a certain structure on an absolute Galois group which makes it easier to study. We are particularly interested in the question of when a field admits a non-trivial parameter-free definable henselian valuation. By a result of Prestel and Ziegler, this does not hold for every henselian valued field. However, improving a result by Koenigsmann, we show that there is a non-trivial parameter-free definable valuation on every henselian valued field. This allows us to give a range of conditions under which a henselian field does indeed admit a non-trivial parameter-free definable henselian valuation. Most of these conditions are in fact of a Galois-theoretic nature. Throughout the thesis, we discuss a number of applications of our results. These include fields elementarily characterized by their absolute Galois group, model complete henselian fields and henselian NIP fields of positive characteristic, as well as PAC and hilbertian fields.

Description logics (DLs) are knowledge representation formalisms with well-understood model-theoretic semantics and computational properties. The DL SROIQ provides the logical underpinning for the semantic web language OWL 2, which is quickly becoming the standard for knowledge representation on the web. A central component of most DL applications is an efficient and scalable reasoner, which provides services such as consistency testing and classification. Despite major advances in DL reasoning algorithms over the last decade, however, ontologies are still encountered in practice that cannot be handled by existing DL reasoners. We present a novel reasoning calculus for the description logic SROIQ which addresses two of the major sources of inefficiency present in the tableau-based reasoning calculi used in state-of-the-art reasoners: unnecessary nondeterminism and unnecessarily large model sizes. Further, we describe a new approach to classification which exploits partial information about the subsumption relation between concept names to reduce both the number of individual subsumption tests performed and the cost of working with large ontologies; our algorithm is applicable to the general problem of deducing a quasi-ordering from a sequence of binary comparisons. We also present techniques for extracting partial information about the subsumption relation from the models generated by constructive DL reasoning methods, such as our hypertableau calculus. Empirical results from a prototypical implementation demonstrate substantial performance improvements compared to existing algorithms and implementations.

In recent years the field of real numbers expanded by a multiplicative subgroup has been studied extensively. In this thesis, the known results will be extended to expansions of the real field. I will consider the structure R consisting of the field of real numbers and an irrational power function. Using Schanuel conditions, I will give a first-order axiomatization of expansions of R by a dense multiplicative subgroup which is a subset of the real algebraic numbers. It will be shown that every definable set in such a structure is a boolean combination of existentially definable sets and that these structures have o-minimal open core. A proof will be given that the Schanuel conditions used in proving these statements hold for co-countably many real numbers. The results mentioned above will also be established for expansions of R by dense multiplicative subgroups which are closed under all power functions definable in R. In this case the results hold under the assumption that the Conjecture on intersection with tori is true. Finally, the structure consisting of R and the discrete multiplicative subgroup 2^{Z} will be analyzed. It will be shown that this structure is not model complete. Further I develop a connection between the theory of Diophantine approximation and this structure.