<p> Starting in the 1970s, Elliot&rsquo;s classification of AF -algebras and Brown-Douglas-Fillmore&rsquo;s classification of essentially normal operators created an explosion in the use of topological methods in the study of C * -algebras. Kasparov&rsquo;s introduction of KK-theory introduced more advanced machinery. This led to better existence and uniqueness theorems with applications in the classification program. In this thesis, I present such a uniqueness theorem with a proof as presented by Eilers-Dadarlat.</p>

<p>We define a ring R to be right cP-Baer if the right annihilator of a cyclic projective right R-module is generated by an idempotent. We also define a ring R to be right I-extending if each ideal generated by an idempotent is right essential in a direct summand of R. It is shown that the two conditions are equivalent in a semiprime ring. Next we define a right I-prime ring, which generalizes the prime condition. This condition is equivalent to all cyclic projective right R-modules being faithful. For a semiprime ring, we show the existence of a cP-Baer hull. We also provide some results about the p.q.-Baer hull and when it is equal to the cP-Baer hull. Polynomial and formal power series rings are studied with respect to the right cP-Baer condition. In general, a formal power series ring over one indeterminate in which its base ring is right p.q.-Baer ring is not necessarily right p.q.-Baer. However, if the base ring is right cP-Baer then the formal power series ring over one indeterminate is right cP-Baer. The fifth chapter is devoted to matrix extensions of right cP-Baer rings. A characterization of when a 2-by-2 generalized upper triangular matrix ring is right cP-Baer is given. The last major theorem is a decomposition of a cP-Baer ring, satisfying a finiteness condition, into a generalized triangular matrix ring with right I-prime rings down the main diagonal. Examples illustrating and delimiting our results are provided.

<p>The family of algebraic surfaces X defined by the single equation [special characters omitted] over an algebraically closed field <i>k</i> of characteristic zero, where a₁, &hellip;, a_n are distinct, is studied. It is shown that this is a rational surface with a non-rational singularity at the origin. The ideal class group of the surface is computed. The terms of the Chase-Harrison-Rosenberg seven term exact sequence on the open complement of the ramification locus of <i>X</i>&rarr;[special characters omitted] are computed; the Brauer group is also studied in this unramified setting.</p><p> The analysis is extended to the surface <i>X&tilde;</i> obtained by blowing up <i>X</i> at the origin. The interplay between properties of <i>X&tilde;</i> , determined in part by the exceptional curve <i> E</i> lying over the origin, and the properties of <i>X</i> is explored. In particular, the implications that these properties have on the Picard group of the surface <i>X</i> are studied.</p>

<p> The study of factorization in integral domains has a long history. Unique factorization domains, like the integers, have been studied extensively for many years. More recently, mathematicians have turned their attention to generalizations of this such as Dedekind domains or other domains which have weaker factorization properties. Many authors have sought to generalize the notion of factorization in domains. One particular method which has encapsulated many of the generalizations into a single study is that of &tau;-factorization, studied extensively by A. Frazier and D.D. Anderson. </p><p> Another generalization comes in the form of studying factorization in rings with zero-divisors. Factorization gets quite complicated when zero-divisors are present due to the existence of several types of associate relations as well as several choices about what to consider the irreducible elements. </p><p> In this thesis, we investigate several methods for extending the theory of &tau;-factorization into rings with zero-divisors. We investigate several methods including: 1) the approach used by A.G. Ag&caron;arg&uuml;n and D.D. Anderson, S. Chun and S. Valdes-Leon in several papers; 2) the method of U-factorization developed by C.R. Fletcher and extended by M. Axtell, J. Stickles, and N. Baeth and 3) the method of regular factorizations and 4) the method of complete factorizations. </p><p> This thesis synthesizes the work done in the theory of generalized factorization and factorization in rings with zero-divisors. Along the way, we encounter several nice applications of the factorization theory. Using &tau;<i>_z</i>-factorizations, we discover a nice relationship with zero-divisor graphs studied by I. Beck as well as D.D. Anderson, D.F. Anderson, A. Frazier, A. Lauve, and P. Livingston. Using &tau;-U-factorization, we are able to answer many questions that arise when discussing direct products of rings. </p><p> There are several benefits to the regular factorization factorization approach due to the various notions of associate and irreducible coinciding on regular elements greatly simplifying many of the finite factorization property relationships. Complete factorization is a very natural and effective approach taken to studying factorization in rings with zero-divisors. There are several nice results stemming from extending &tau;-factorization in this way. Lastly, an appendix is provided in which several examples of rings satisfying the various finite factorization properties studied throughout the thesis are given.</p>

<p> In 1988, Mayer published a strong form of Vaught's Conjecture for o-minimal theories. She showed Vaught's Conjecture holds, and characterized the number of countable models of an o-minimal theory <i>T</i> if <i> T</i> has fewer than continuum many countable models. Friedman and Stanley have shown that several elementary classes are Borel complete. In this thesis we address the class of countable models of an o-minimal theory <i>T </i> when <i>T</i> has continuum many countable models. Our main result gives a model theoretic dichotomy describing the Borel complexity of isomorphism on the class of countable models of <i>T</i>. The first case is if <i>T</i> has no simple types, isomorphism is Borel on the class of countable models of <i>T</i>. In the second case, <i> T</i> has a simple type over a finite set <i>A</i>, and there is a finite set <i>B</i> containing <i>A</i> such that the class of countable models of the completion of <i>T </i>over <i> B</i> is Borel complete.</p>

<p> We investigate the properties of finite tuples of commuting isometries that are constrained by a system of polynomial equations. More precisely, suppose <i>I</i> is an ideal in the ring of complex <i> n</i>-variable polynomials and that <i>I</i> determines an affine algebraic variety of dimension 1. Further, suppose that there are <i> n</i> commuting Hilbert space isometries <i>V</i>₁, . . . ,<i>V_n</i> with the property that <i>p</i>(<i> V₁</i>, . . . ,<i>Vⁿ</i>) = 0 for each <i>p</i> in the ideal <i>I.</i> Because the <i> n</i>-tuple (<i>V</i>₁, . . . ,<i>V_n</i>) can be decomposed as a direct sum of an <i>n</i>-tuple of unitary operators and a completely non-unitary <i>n</i>-tuple, we assume that the unitary summand is trivial. Under these assumptions, we can decompose the <i>n</i>-tuple as a finite direct sum of <i>n</i>-tuples of the form (<i>T</i>₁, . . . ,<i>T_n</i>), where each <i>T_i</i> either is multiplication by a scalar or is unitarily equivalent to a unilaterial shift of some multiplicity. We then focus on the special case in which <i>V</i>₁, . . . ,<i>V_n</i> are generalized shifts of finite multiplicity. In this case we are able to classify such <i>n</i>-tuples up to something we term &lsquo;virtual similarity&rsquo; using two pieces of data : the ideal of all polynomials p such that <i>p</i>(<i>V</i>₁, . . . ,<i>V_n</i>) = 0 and a finite tuple of positive integers.</p><p>

Cycle Systems

<p> In this work we investigate the global symmetries of six-dimensional superconformal field theories (6D SCFTs) via their description in F-theory. We provide computer algebra system routines determining global symmetry maxima for all known 6D SCFTs while tracking the singularity types of the associated elliptic fibrations. We tabulate these bounds for many CFTs including every 0-link based theory. The approach we take provides explicit tracking of geometric information which has remained implicit in the classifications of 6D SCFTs to date. We derive a variety of new geometric restrictions on collections of singularity collisions in elliptically fibered Calabi-Yau varieties and collect data from local model analyses of these collisions. The resulting restrictions are sufficient to match the known gauge enhancement structure constraints for all 6D SCFTs without appeal to anomaly cancellation and enable our global symmetry computations for F-theory SCFT models to proceed similarly. </p><p>