We study annular algebras associated to a rigid C*-tensor category, a generalization of both Ocneanu's tube algebra and Jones' affine annular category. We show that all ``sufficiently large' annular algebras are strongly morita equivalent, hence have equivalent representation theories. We then demonstrate the existence of a universal C*-algebra for the tube algebra, analogous to the universal C*-algebra of a discrete group. Using this perspective, we show that a ``piece' of the representation theory of the tube algebra is precisely the admissible representation theory of the fusion algebra introduced by Popa and Vaes, allowing for properties such as amenability, the Haagerup property, and property (T) for the category to be interpreted in terms on annular representations. We treat several examples, and partially characterize the C*-algebras associated to the Temperley-Lieb-Jones categories with negative q parameter. As an application of our annular perspective, we show that quantum G_2 categories have property (T) for positive q not equal to 1.

We explore the descriptive set theory of subgroups of fundamental groups, giving theorems regarding dichotomies on cardinality. In paticular, we show that the quotient of the fundamental group of a path connected, locally path connected Polish space by a normal subgroup which is sufficiently eay to describe topologically (of ``nice' pointclass having the property of Baire) is either countable or of cardinality continuum. In case the space is compact, countability of the quotient implies the quotient is finitely generated. We give upper bounds on the complexity of some subgroups, such as the shape kernel and the Spanier group. Applications to the normal generation of groups are given, as well as an application to covering space theory. We present an array of theorems regarding the first homology of Peano continua. We also demonstrate the existence of subgroups of all but finitely many additive and multiplicative Borel types. We prove that torsion-free word hyperbolic groups are n-slender, and the class of n-slender groups is closed under graph products.

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A primitive Pythagorean triple is a set of positive integers, x, y and z, such that x and y are relatively prime and x2+y2=z2. In this thesis several partitions of the set Pp of primitive Pythagorean triples are studied. Also interesting properties of these partitions are derived. These properties are used to develop a two-dimensional array of Pp.

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Reflection positivity has several applications in both mathematics and physics. For example, reflection positivity induces a duality between group representations. In this thesis, we coin a new definition for a new kind of reflection positivity, namely, twisted reflection positive representation on a vector space. We show that all of the non-compactly causal symmetric spaces give rise to twisted reflection positive representations. We discover examples of twisted reflection positive representations on the sphere and on the Grassmannian manifold which are not unitary, namely, the generalized principle series with the Cosine transform as an intertwining operator. We give a direct proof for the reflection positivity of the Cosine transform on SO(n). On the other hand, we generalize an integrability theorem to the case of non-positive definite distribution. As a result, we give a relation between the non-compactly causal symmetric spaces and the reflection positive distributions. Cocycle conditions are also treated. We construct a general method to generate twisted reflection positive representations and then we apply it to get twisted reflection positive representations on the Euclidean space. Finally, we introduce a reflection positive cyclic distribution vector for the circle case. Then we prove that this distribution vector generates a well known reflection positive function.

<p>We introduce a new model to study the behavior of a portfolio of defaultable assets. We refer to this model as the Gaussian-Poisson model. It builds upon one-factor Gaussian copula models and Poisson models (specifically Cox processes). Our model utilizes a random variable <i>Y</i> along with probability measures ℙ and ℙ. The measures ℙ and ℙ will act as market pricing measures and are obtained via conditioning. The random variable <i>Y</i> will act as a default descriptor.</p> <p> We provide the distribution of <i>Y</i> under both ℙ and ℙ. We use a conditional probability to examine expected portfolio and tranche losses, with applications including credit default swaps and collateralized debt obligations. The Gaussian-Poisson model requires a choice of an intensity model. We examine a portfolio loss' dependence upon parameters of the intensity model. Finally, we present three possible models of the intensity process. </p>

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