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Brauer-Picard Groups And Pointed Braided Tensor Categories, Costel Gabriel Bontea

Doctoral Dissertations

Tensor categories are ubiquitous in areas of mathematics involving algebraic structures. They appear, also, in other fields, such as mathematical physics (conformal field theory) and theoretical computer science (quantum computation). The study of tensor categories is, thus, a useful undertaking.

Two classes of tensor categories arise naturally in this study. One consists of group-graded extensions and another of pointed tensor categories. Understanding the former involves knowledge of the Brauer-Picard group of a tensor category, while results about pointed Hopf algebras provide insights into the structure of the latter.

This work consists of two main parts. In the first one we …

A Dynamical-Systems Approach To Understanding Turbulence In Plane Couette Flow, Mimi Szeto

Doctoral Dissertations

Dynamical systems theory is used to understand the dynamics of low-dimensional spatio-temporal chaos. Our research aimed to apply the theory to understanding turbulent fluid flows, which could be thought of as spatio-temporal chaos in a very-high dimensional space. The theory explains a system's dynamics in terms of the local dynamics of its periodic solutions; these are the periodic orbits in state space. We considered the development of a model for the dynamics of plane Couette flow based on the theory. The proposed model is essentially a set of low-dimensional models for the local dynamics of the periodic orbits of the …

A Beurling Theorem For Noncommutative Hardy Spaces Associated With A Semifinite Von Neumann Algebra With Various Norms, Lauren Beth Meitzler Sager

Doctoral Dissertations

We prove Beurling-type theorems for H-invariant spaces in relation to a semifinite von Neu-mann algebra M with a semifinite, faithful, normal tracial weight τ, using an extension of Arveson’s non-commutative Hardy space H-. First we prove a Beurling-Blecher-Labuschagne theorem for H-invariant subspaces of L p (M,τ) when 0 < p ≤ -. We also prove a Beurling-Chen-Hadwin-Shen theorem for H -invariant subspaces of L a (M,τ) where a is a unitarily invariant, locally k 1 -dominating, mutually continuous norm with respect to &\tau;. For a crossed product of a von Neumann algebra M by an action β, M o β Z, we are able to completely characterize all H-invariant subspaces of L a (Mo β Z,t) using our results. As an example, we completely characterize all H-invariant subspaces of the Schatten p-class, S p (H) (0 < p ≤ -), where H - is the lower tri-angular subalgebra of B(H). We also characterize the non-commutative Hardy space H -invariant subspaces in a Banach function space I(τ) on a semifinite von Neumann algebra M.

On Tensor Autoequivalences Of Graded Fusion Categories, Ian Swenson Marshall

Doctoral Dissertations

Fusion categories generalize the representation theory of finite groups. The simplest examples of fusion categories come from finite groups, their representations, and their cohomology. In general, it is useful to examine group theoretical features of fusion categories such as groups of (isomorphism classes of) tensor invertible objects, and gradings by finite groups. Indeed, every fusion category has a maximal pointed subcategory (generated by tensor invertible objects) and a universal grading by a finite group. We use such features to study tensor autoequivalences.

Pointed fusion categories: categories for which all simple objects are tensor invertible, provide our prototype for graded fusion …

Porous Medium Convection At Large Rayleigh Number: Studies Of Coherent Structure, Transport, And Reduced Dynamics, Baole Wen

Doctoral Dissertations

Buoyancy-driven convection in fluid-saturated porous media is a key environmental and technological process, with applications ranging from carbon dioxide storage in terrestrial aquifers to the design of compact heat exchangers. Porous medium convection is also a paradigm for forced-dissipative infinite-dimensional dynamical systems, exhibiting spatiotemporally chaotic dynamics if not ``true" turbulence. The objective of this dissertation research is to quantitatively characterize the dynamics and heat transport in two-dimensional horizontal and inclined porous medium convection between isothermal plane parallel boundaries at asymptotically large values of the Rayleigh number $Ra$ by investigating the emergent, quasi-coherent flow. This investigation employs a complement of direct …

What Do Students Do In Self-Formed Mathematics Study Groups?, Gillian E. Galle

Doctoral Dissertations

An implicit assumption of many university classes is that students will spend a large amount of time outside the classroom refining their understanding of the material to develop mastery of the concepts. This is especially true in first year mathematics courses at the undergraduate level. However, little is known about what students do to fulfill this didactical contract with their instructors. The currently available research relies primarily on self-reported data from the students collected through questionnaires or interviews. This study sought to start describing what students do while studying mathematics in a self-created group outside of the classroom setting through …

The Multifaceted Nature Of Mathematics Knowledge For Teaching: Understanding The Use Of Teachers' Specialized Content Knowledge And The Role Of Teachers' Beliefs From A Practice-Based Perspective, Lauren E. Provost

Doctoral Dissertations

This work investigates middle school teachers' mathematics knowledge for teaching (MKT) as defined by Hill (2007). Within this two-part dissertation, the level of MKT was considered as well as the role of teacher beliefs in actual specialized content knowledge (SCK) use, a specific type of mathematics knowledge for teaching vital in quality mathematics instruction. Additionally, the model of MKT knowledge was explored through confirmatory factor analysis on a large, national dataset of middle school mathematics teacher survey responses involving mathematics knowledge for teaching. SCK was found to be vital in quality mathematics instruction yet not sufficient. Teacher beliefs about the …

On Decompositions And Connes's Embedding Problem Of Finite Von Neumann Algebras, Jinsong Wu

Doctoral Dissertations

A longstanding open question of Connes asks whether every finite von Neumann algebra embeds into an ultraproduct of finite-dimensional matrix algebras. As of yet, algebras verified to satisfy Connes's embedding property belong to just a few special classes (e.g. amenable algebras and free group factors). In this dissertation we establish Connes's embedding property for von Neumann algebras satisfying Popa's co-amenability condition. Some decomposition properties of finite von Neumann algebras are also investigated.

Chapter 1 reviews von Neumann algebras, completely bounded mappings, conditional expectations, tensor products, crossed products, direct integrals, and Jones basic construction.

Chapter 2 introduces new decompositions of finite …

Proof And Reasoning In An Inquiry-Oriented Class: The Impact Of Classroom Discourse, Susan D. Generazzo

Doctoral Dissertations

Over the past decade, mathematics educators and researchers have become increasingly aware of the impact of social interactions on students' learning (NCTM, 2000; Bowers & Nickerson, 2001; Forman, 2003). Current research indicates that the classroom environment, including the activities and discussions that take place, can have a significant effect on the ways students make sense of mathematical concepts (Yackel, 2001). Understanding mathematics involves knowing how to make sense of key concepts through the processes of reasoning and justification. Educators and researchers agree on the importance of providing students with opportunities in class to explore, conjecture, and prove in order to …

Homotopy Mapping Spaces, Jeremy Brazas

Doctoral Dissertations

In algebraic topology, one studies the group structure of sets of homotopy classes of maps (such as the homotopy groups pin( X)) to obtain information about the spaces in question. It is also possible to place natural topologies on these groups that remember local properties ignored by the algebraic structure. Upon choosing a topology, one is left to wonder how well the added topological structure interacts with the group structure and which results in homotopy theory admit topological analogues. A natural place to begin is to view the n-th homotopy group pi n(X) as the quotient space of the iterated …

Models And Methods For Computationally Efficient Analysis Of Large Spatial And Spatio-Temporal Data, Chengwei Yuan

Doctoral Dissertations

With the development of technology, massive amounts of data are often observed at a large number of spatial locations (n). However, statistical analysis is usually not feasible or not computationally efficient for such large dataset. This is the so-called "big n problem".

The goal of this dissertation is to contribute solutions to the "big n problem". The dissertation is devoted to computationally efficient methods and models for large spatial and spatio-temporal data. Several approximation methods to "the big n problem" are reviewed, and an extended autoregressive model, called the EAR model, is proposed as a parsimonious model that accounts for …

Bimodule Categories And Monoidal 2-Structure, Justin Greenough

Doctoral Dissertations

We define a notion of tensor product of bimodule categories and prove that with this product the 2-category of C -bimodule categories for fixed tensor C is a monoidal 2-category in the sense of Kapranov and Voevodsky ([KV91]). We then provide a monoidal-structure preserving 2-equivalence between the 2-category of C -bimodule categories and Z( C )-module categories (module categories over the center of C ). The (braided) tensor structure of C1⊠D C2 for (braided) fusion categories over braided fusion D is introduced. For a finite group G we show that de-equivariantization is equivalent to the tensor product over Rep( G). …

Conjecturing In Dynamic Geometry: A Model For Conjecture-Generation Through Maintaining Dragging, Anna Baccaglini-Frank

Doctoral Dissertations

The purpose of this research is to study aspects of the impact of Dynamic Geometry Systems (DGS) in the process of producing conjectures in Euclidean geometry. Previous research has identified and classified a set of dragging schemes spontaneously used by students. Building on these findings, the study focuses on cognitive processes that arise in correspondence to particular dragging modalities in Cabri. Specifically, we have conceived a model describing what seems to occur during a process of conjecture-generation that involves the use of a particular dragging modality, described in the literature as dummy locus dragging. In order to accomplish this goal, …

The Impact Of A Mathematics Research Experience On Teachers' Conceptions Of Student Learning, Todd Abel

Doctoral Dissertations

Many mathematics teacher professional development programs have either incorporated or been organized around a goal of providing "research-like" (Cuoco, 2001) experiences. That is, teachers participate in a project that somehow simulates the mathematics research process. Though some research studies have shown positive outcomes from such programs, researchers have cautioned against assuming universally positive benefits without sufficient evidence (Proulx and Bednarz, 2001). Teacher conceptions of student learning play an important role in lesson development and preparation for classroom work (Penso & Shoham, 2003). Similarities between the processes of mathematics research and student learning (Dreyfus, 1991) beg the question of whether experience …

Admissible Orders On Quotients Of The Free Associative Algebra, Jeremiah William Johnson

Doctoral Dissertations

An admissible order on a multiplicative basis of a noncommutative algebra A is a term order satisfying additional conditions that allow for the construction of Grobner bases for A -modules. When A is commutative, a finite reduced Grobner basis for an A -module can always be obtained, but when A is not commutative this is not the case; in fact in many cases a Grobner basis theory for A may not even exist.

E. Hinson has used position-dependent weights, encoded in so-called admissible arrays, to partially order words in the free associative algebra in a way which produces a length-dominant …

Mf Algebras And A Bishop -Stone -Weierstrass Theorem Result, Qihui Li

Doctoral Dissertations

This dissertation consists of two parts. In the first part, we obtain many new results about MF algebras. First, we continue the work on D. Voiculescu's topological free entropy dimension deltatop (x1, ..., xn) for an n-tuple x&ar; = (x1, ..., xn) of elements in a unital C*-algebra. We also introduce a new invariant that is a C*-algebra analog of the invariant K3 introduced for von Neumann algebras. Second, we discuss a full amalgamated free product of unital MF (and residually finite-dimensional) algebras with amalgamation over a finite-dimensional C*-subalgebra. Necessary and sufficient conditions are given in this situation. In the …

Von Neumann Algebras, Affiliated Operators And Representations Of The Heisenberg Relation, Zhe Liu

Doctoral Dissertations

Von Neumann algebras are self-adjoint, strong-operator closed subalgebras (containing the identity operator) of the algebra of all bounded operators on a Hilbert space. Factors are von Neumann algebras whose centers consist of scalar multiples of the identity operator. In this thesis, we study unbounded operators affiliated with finite von Neumann algebras, in particular, factors of Type II1. Such unbounded operators permit all the formal algebraic manipulations used by the founders of quantum mechanics in their mathematical formulation, and surprisingly, they form an algebra. The operators affiliated with an infinite von Neumann algebra never form such an algebra. The Heisenberg commutation …

Invariant Frechet Algebras On Bounded Symmetric Domains, Oleg Eroshkin

Doctoral Dissertations

Let D be a bounded domain in the complex vector space Cn . We say that D is symmetric iff, given any two points p, q ∈ D, there is a biholomorphism &phis;, which interchanges p and q. These domains were classified abstractly by Elie Cartan in his general study of symmetric spaces, and were canonically realized in Cn by Harish-Chandra. They include polydisks and Siegel domains.

Let D be a bounded symmetric domain in Cn , and G be the largest connected group of biholomorphic automorphisms of D. The algebra C( D) of all continuous (not necessarily bounded) complex-valued …

Wavelet Regression With Long Memory Infinite Moving Average Errors, Juan Liu

Doctoral Dissertations

For more than a decade there has been great interest in wavelets and wavelet-based methods. Among the most successful applications of wavelets is nonparametric statistical estimation, following the pioneering work of Donoho and Johnstone (1994, 1995) and Donoho et al. (1995). In this thesis, we consider the wavelet-based estimators of the mean regression function with long memory infinite moving average errors, and investigate the rates of convergence of estimators based on thresholding of empirical wavelet coefficients. We show that these estimators achieve nearly optimal minimax convergence rates within a logarithmic term over a large class of non-smooth functions that involve …

Kadison -Singer Algebras With Applications To Von Neumann Algebras, Mohan Ravichandran

Doctoral Dissertations

I develop the theory of Kadison-Singer algebras, introduced recently by Ge and Yuan. I prove basic structure theorems, construct several new examples and explore connections to other areas of operator algebras. In chapter 1, I survey those aspects of the theory of non-selfadjoint algebras that are relevant to this work. In chapter 2, I define Kadison-Singer algebras and give different proofs of results of Ge-Yuan, which will be further extended in the last chapter. In chapter 3, I analyse in detail a class of elementary Kadison-Singer algebras that contain Hinfinity and describe their lattices of projections. In chapter 4, I …

Mathematics Of Double-Walled Carbon Nanotube Model: Asymptotic Spectral And Stability Analysis, Miriam Rojas-Arenaza

Doctoral Dissertations

This dissertation is devoted to analytical study of a contemporary model of a double-walled carbon nano-tube. Carbon nano-tubes have been considered outstanding candidates to innovate and to promote emerging technologies, due to their remarkable chemical, mechanical, and physical properties. For these technologies, there is a need to develop mathematical models that capture the nature of the responses of these structures under a variety of physical conditions. Developing these models is challenging because the behavior lies on the borderline between classical and quantum systems. The main goal of the present dissertation is to prove mathematically rigorous results concerning the vibrational behavior …

The Process Of Making Meaning: The Interplay Between Teachers' Knowledge Of Mathematical Proofs And Their Classroom Practices, Megan Paddack

Doctoral Dissertations

The purpose of this study was to investigate and describe how middle school mathematics teachers make meaning of proofs and the process of proving in the context of their classroom practices. A framework of making meaning, created by the researcher, guided the data collection and analysis phases of the study. This framework describes the five central aspects of the process of making meaning: knowledge, beliefs, utilization of knowledge, interconnections of practice and knowledge, and making sense of past knowledge and current experiences. The utilization of a qualitative research methodology that combined ethnographic fieldwork and discourse analysis allowed the researcher to …

Kadison -Singer Algebras, Wei Yuan

Doctoral Dissertations

In this dissertation, we defined a new class of non selfadjoint operator algebras---Kadison-Singer algebras or KS-algebras for simplicity. These algebras combine triangularity, reflexivity and von Neumann algebra property into one consideration. Generally speaking, KS-algebras are reflexive, maximal triangular with respect to its "diagonal subalgebra". Many selfadjoint features are preserved in them and concepts can be borrowed directly from the theory of von Neumann algebras. In fact, a more direct connection of KS-algebras and von Neumann algebras is through the lattice of invariant projections of a KS-algebra. The lattice is reflexive and "minimally generating" in the sense that it generates the …

Unitarily Invariant Norms And Tensor Products Of Maximal Injective Von Neumann Subalgebras, Junsheng Fang

Doctoral Dissertations

This dissertation consists of four contributions to the study of von Neumann algebras. In the first part, we set up a representation theorem for unitarily invariant norms on finite factor von Neumann algebras. In the second part, we set up a representation theorem for unitarily invariant norms related to infinite factor von Neumann algebras. In the third part, we introduce a notion of completely singular von Neumann subalgebras and characterize the class of completely singular von Neumann subalgebras. In the fourth part, we study a longstanding open question on the tensor product of maximal injective von Neumann subalgebras.

The first …

Planar Algebra Of Families Of Subfactors, Paramita Das

Doctoral Dissertations

The standard invariant is an object that classifies a fairly general class of subfactors and was reformulated by Vaughan Jones as 'planar algebras'. We compute the planar algebra of two families of subfactors arising out of actions of finitely generated groups on a II 1 factor in an attempt to see whether this class can be enlarged. This is a joint work with Dietmar Bisch and Shamindra Ghosh.

Free Entropy Dimensions And Approximate Liftings, Weihua Li

Doctoral Dissertations

In the first chapter of the dissertation, we give a very elementary proof of a more detailed version of one of D. Voiculescu's results, which was a key ingredient in Voiculescu's proof that his free entropy is additive when the variables are free.

In the second chapter of the dissertation, based on the notion of upper free orbit-dimension introduced by D. Hadwin and J. Shen, we introduce a new invariant on finite von Neumann algebras that do not necessarily act on separable Hilbert space. We show that this invariant is independent of the generating set, and we obtain a number …

Understanding Abstract Algebra Concepts, Anna S. Titova

Doctoral Dissertations

This study discusses various theoretical perspectives on abstract concept formation. Students' reasoning about abstract objects is described based on theoretical proposition that abstraction is a shift from abstract to concrete. Existing literature suggested a theoretical framework for the study. The framework describes process of abstraction through its elements: assembling, theoretical generalization into abstract entity, and articulation. The elements of the theoretical framework are identified from students' interpretations of and manipulations with elementary abstract algebra concepts including the concepts of binary operation, identity and inverse element, group, subgroup, cyclic group. To accomplish this, students participating in the abstract algebra class were …

Morita Equivalence For Group -Theoretical Categories, Deepak Naidu

Doctoral Dissertations

We give necessary and sufficient conditions for two pointed categories to be dual to each other with respect to a module category. Whenever the dual of a pointed category with respect to a module category is pointed, we give explicit formulas for the Grothendieck ring and for the associator of the dual. This leads to the definition of categorical Morita equivalence on the set of all finite groups and on the set of all pairs ( G, o), where G is a finite group and o ∈ H3(G, kx). A group-theoretical and cohomological interpretation of this relation is given. As …

Planar Algebras: A Category Theoretic Point Of View, Shamindra Kumar Ghosh

Doctoral Dissertations

We define Jones's planar algebra as a map of multicategories and construct a planar algebra starting from a 1-cell in a pivotal strict 2-category. We introduce the concept of affine representations of a planar algebra and prove some finiteness results for the affine representations of finite depth planar algebras. We also show that the radius of convergence of the dimension of an affine representation of the planar algebra associated to a finite depth subfactor is at least as big as the inverse-square of the modulus.

An Investigation Of Pre-Service Teachers' And Professional Mathematicians' Perceptions Of Mathematical Proof At The Secondary School Level, David M. Gray

Doctoral Dissertations

The National Council of Teachers of Mathematics [NCTM] states that by the time students graduate high school, they should learn to present "arguments consisting of logically rigorous deductions of conclusions from hypotheses" (NCTM, 2000, p. 56) in "written forms that would be acceptable to professional mathematicians" (NCTM, 2000, p. 58). Research studies indicate, however, that students and teachers have difficulty with many aspects of mathematical proof, including its nature and meaning. In addition, there appears to be a disconnect between school teachers' and university mathematicians' expectations for their respective students regarding mathematical proof. This study examined the perceptions of thirteen …