Quandles are distributive algebraic structures originally introduced independently by David Joyce and Sergei Matveev in 1979, motivated by the study of knots. In this dissertation, we discuss a number of generalizations of the notion of quandles. In the first part of this dissertation we discuss biquandles, in the context of augmented biquandles, a representation of biquandles in terms of actions of a set by an augmentation group. Using this representation we are able to develop a homology and cohomology theory for these structures. We then introduce an n-ary generalization of the notion of quandles. We discuss a number of properties of these structures and provide a number of examples. Also discussed are methods of obtaining n-ary quandles through iteration of binary quandles, and obtaining binary quandles from n-ary quandles, along with a classification of low order ternary quandles. We build upon this generalization, introducing n-ary f-quandles, and similarly discuss examples, properties, and relations between the n-ary structures and their binary counter parts, as well as low order classification of ternary f-quandles. Finally we present cohomology theory for general n-ary f-quandles.

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For an inclusion N \subseteq M of finite von Neumann algebras, we study the group of normalizers N_M(B) = {u: uBu^* = B} and the von Neumann algebra it generates. In the first part of the dissertation, we focus on the special case in which N \subseteq M is an inclusion of separable II_1 factors. We show that N_M(B) imposes a certain "discrete" structure on the generated von Neumann algebra. By analyzing the bimodule structure of certain subalgebras of N_M(B)'', this leads to a "Galois-type" theorem for normalizers, in which we find a description of the subalgebras of N_M(B)'' in terms of a unique countable subgroup of N_M(B). We then apply these general techniques to obtain results for inclusions B \subseteq M arising from the crossed product, group von Neumann algebra, and tensor product constructions. Our work also leads to a construction of new examples of norming subalgebras in finite von Neumann algebras: If N \subseteq M is a regular inclusion of II_1 factors, then N norms M: These new results and techniques develop further the study of normalizers of subfactors of II_1 factors. The second part of the dissertation is devoted to studying normalizers of maximal abelian self-adjoint subalgebras (masas) in nonseparable II_1 factors. We obtain a characterization of masas in separable II_1 subfactors of nonseparable II_1 factors, with a view toward computing cohomology groups. We prove that for a type II_1 factor N with a Cartan masa, the Hochschild cohomology groups H^n(N,N)=0, for all n \geq 1. This generalizes the result of Sinclair and Smith, who proved this for all N having separable predual.

Quasicrystal is the term used for a solid that possesses an essentially discrete diffraction pattern without having translational symmetry. Compared to periodic crystals, this difference in structure gives quasicrystals new properties that make them interesting to study -- both from a mathematical and from a physical point of view. In this thesis we review a mathematical description of quasicrystals that aims at generalizing the well-established theory of periodic crystals. We see how this theory can be connected to the cohomology of groups and how we can use this connection to classify quasicrystals. We also review an experimental method, NIXSW (Normal Incidence X-ray Standing Waves), that is ordinarily used for surface structure determination of periodic crystals, and show how it can be used in the study of quasicrystal surfaces. Finally, we define the reduced lattice and show a way to plot lattices in MATLAB. We see that there is a connection between the diffraction pattern and the reduced lattice and we suggest a way to describe this connection.

For T the circle group, we construct a differential refinement of T-equivariant K-theory. We first construct a de Rham model for delocalized equivariant cohomology and a delocalized equivariant Chern character based on [19] and [14]. We show that the delocalized equivariant Chern character induces a complex isomorphism. We then construct a geometric model for differential T-equivariant K-theory analogous to the model of differential K-theory in [27] and deduce its basic properties. / text

The construction of characteristic classes via the curvature form of a connection is one motivation for the refinement of integral cohomology by de Rham cocycles -- known as differential cohomology. We will discuss the analog in the case of a group action on the manifold: We will show the compatibility of the equivariant characteristic class in integral Borel cohomology with the equivariant characteristic form in the Cartan model. Motivated by this understanding of characteristic forms, we define equivariant differential cohomology as a refinement of equivariant integral cohomology by Cartan cocycles.

The algebra of matrices M with entries in an abelian von Neumann algebra is a C*-module. C*-modules were originally defined and studied by Kaplansky and we outline the foundations of the theory and particular properties of M. Furthermore, we prove a structure theorem for ultraweakly closed submodules of M, using techniques from the theory of type I finite von Neumann algebras. By analogy with the classical join in topology, the join for operator algebras A and B acting on Hilbert spaces H and K, respectively, was defined by Gilfeather and Smith. Assuming that K is finite dimensional, Gilfeather and Smith calculated the Hochschild cohomology groups of the join. We assume that M is the algebra of matrices with entries in a maximal abelian von Neumann algebra U, A is an operator algebra acting on a Hilbert space K, and B is an ultraweakly closed subalgebra of M containing U. In this new context, we redefine the join, generalize the calculations of Gilfeather and Smith, and calculate the cohomology groups of the join.

Thesis (Ph. D.)--Duke University, 2007. / Includes bibliographical references.

Originally presented as the author's Thesis (Ph. D.)--Scuola normale superiore Pisa. / Includes bibliographical references (p. [125]-131) and index.