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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.