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The author of this dissertation studies the spectral properties of high-contrast photonic crystals, i.e. periodic electromagnetic waveguides made of two materials (a connected phase and included phase) whose electromagnetic material properties are in large contrast. A spectral analysis of 2nd-order divergence-form partial differential operators (with a coupling constant k) is provided. A result of this analysis is a uniformly convergent power series representation of Bloch-wave eigenvalues in terms of the coupling constant k in the high-contrast limit k -> infinity. An explicit radius of convergence for this power series is obtained, and can be written explicitly in terms of the Bloch-wave vector, the Dirichlet eigenvalues of the inclusion geometry, and a lower bound on another spectrum known as the " generalized electrostatic resonances " . This lower bound is derived from geometric properties of the inclusion geometry for the photonic crystal.

The goal of this dissertation is to expand Berhard Koopman's operator theoretic global linearization approach to the study of nonautonomous flows. Given a system with states x in a set \Omega (the state space), a map t\to \gamma(t,s,x) (t\geq s \geq 0) is called a global flow if it describes the time evolution of a system with the initial state x \in \Omega at time t \geq s \geq 0. Koopman's approach to the study of flows is to look at the dynamics of the observables of the states instead of studying the dynamics of the states directly. To do so, one considers a vector space Z containing observables (measurements) and a vector space \mathcal{M}:=\mathcal{F}([0,\infty)\times \Omega,Z) of functions containing observations g: [0,\infty) \times \Omega \to Z. Then every global flow \gamma induces a family T(t)(t\geq 0) of linear maps on \mathcal{M}, where \begin{equation}\label{abstract} T(t)g: (s,x) \mapsto g(t+s,\gamma(t+s,s,x)). \end{equation} Since every global flow \gamma satisfies \gamma(s,s,x) = x and \gamma(t,r,\gamma(r,s,x)) = \gamma(t,s,x) for t\geq r \geq s \geq 0 and x \in \Omega, the linear maps T(t)(t\geq 0) define an operator semigroup on \mathcal{M}; that is, T(0) = I \text and T(t+s) = T(t)T(s) for t,s \geq 0. Following Koopman's approach, in pursuit of understanding the flow \gamma, we investigate the linear flow semigroup T(t)(t\geq 0) on \M given by \eqref{abstract}, and if \gamma(t,s,x) = U(t,s)x for some linear evolution family U(t,s), an associated special evolution semigroups on a subspace of \M given by S(t)f: s\mapsto f(t+s)U(t+s,s). Of primary concern are continuity properties of the associated linear evolution semigroups on different function spaces (Chapters 1-3). The Lie generator of the flow and a collection of open problems concerning general flow semigroups \eqref{abstract}, asymptotics and/or finite time blow-up, and Lie-Totter type approximations are described in Chapter 4.

Certain universal features of photonic resonant scattering systems are encapsulated in a simple model which is a resonant modification of the famous Lamb Model for free vibrations of a nucleus in an extended medium. We analyze this "resonant Lamb model" to garner information on dynamic resonant scattering of near-monochromatic fields when an extended system is weakly coupled to a resonator. The transmitted field in a resonant scattering process consists of two distinct pathways: an initial pulse (direct transmission) and a tail of slow decay (resonant transmission). The resonant Lamb model incorporates a two-part scatterer attached to an infinite string with a continuous spectrum. The non-resonant part of the scatterer is associated with direct scattering; and the resonant part is associated with field amplification and delayed transmission. We provide a mathematical characterization of the "direct transmission" and the "resonant transmission" by analyzing the pole structure of the resolvent operator of the system. The coupling constant (gamma), the proximity of resonance to the central frequency of incidence (eta) and the spectral width (sigma) of the incident pulse are three distinguished parameters that are small and affect resonance in the high-Q and near-monochromatic regime. The main objectives of this work are to analyze resonant amplification and transmission anomalies in the simultaneous High-Q and near-monochromatic regime as they depend on the three aforementioned parameters and to quantify the accuracy of coupled mode theory in that same regime.

In this work on Polynomial Identity (PI) quantized Weyl algebras we begin with a brief survey of Poisson geometry and quantum cluster algebras, before using these as tools to classify the possible centers of such algebras in two different ways. In doing so we explicitly calculate the formulas of the discriminants of these algebras in terms of a general class of central polynomial subalgebras. From this we can classify all members of this family of algebras free over their centers while proving that their discriminants have the properties of effectiveness and local domination. Applying these results to the family of tensor products of PI quantized Weyl algebras we solve the automorphism and isomorphism problems.

A topological space X is compact-finite if and only if compactness and finiteness are equivalent. The most commonly used term for such a space is cf. CF-spaces may be determined in many ways. However, to show that a space is cf, it suffices to prove that every compact set is finite or that every infinite set is not compact. Numerous examples and related theorems of cf-spaces are presented.

Amenability is a fundamental in operator algebras. The classification of von Neumann algebras by Alain Connes is a milestone in the theory. The study of amenable subalgebras in II1 factors has led to many important developments such as the computation of the fundamental groups, strong solidity of free group factors, etc. In this thesis we consider a question about amenable extension in II1 factors, namely, given a diffuse amenable subalgebra in a II1 factor, in how many ways it can be extended to some maximal amenable subalgebra? We give two classes of examples where unique amenable extension results are obtained. The key notion we use is a strengthening of Popaâs asymptotic orthogonality property.

<p> The main problem we consider in this thesis is the essential self-adjointness of the symplectic Dirac operators D and D constructed by Katharina Habermann in the mid 1990s. Her constructions run parallel to those of the well-known Riemannian Dirac operators, and show that in the symplectic setting many of the same properties hold. For example, the symplectic Dirac operators are also unbounded and symmetric, as in the Riemannian case, with one important difference: the bundle of symplectic spinors is now infinite-dimensional, and in fact a Hilbert bundle. This infinite dimensionality makes the classical proofs of essential self-adjointness fail at a crucial step, namely in local coordinates the coefficients are now seen to be unbounded operators on L2(Rn). A new approach is needed, and that is the content of these notes. We use the decomposition of the spinor bundle into countably many finite-dimensional subbundles, the eigenbundles of the harmonic oscillator, along with the simple behavior of D and D with respect to this decomposition, to construct an inductive argument for their essential self-adjointness. This requires the use of ancillary operators, constructed out of the symplectic Dirac operators, whose behavior with respect to the decomposition is transparent. By an analysis of their kernels we manage to deduce the main result one eigensection at a time.</p>

This thesis deals obtaining global a priori estimates for quasilinear elliptic equations and sharp existence results for Quasilinear equations with gradient nonlinearity on the right. The main results are contained in Chapters 3, 4, 5 and 6. In Chapters 3 and 4, we obtain global unweighted a priori estimates for very weak solutions below the natural exponent and weighted estimates at the natural exponent. The weights we consider are the well studied Muckenhoupt weights. Using the results obtained in Chapter 4, we obtain sharp existence result for quasilinear operators with gradient type nonlinearity on the right. We characterize the function space which yields such sharp existence results. Finally in Chapter 6, we prove existence of very weak solutions to quasilinear equations below the natural exponent with measure data on the right.

Let $f in ell^2(I)$ be a signal at time $t = 0$ of an evolution process controlled by a bounded linear operator A that produces the signals $Af, A^2f, cdots $ at times $t=1,2,cdots$. Let $Y ={f(i), Af(i), cdots, A^{l_i}f(i) : i in Omega subset I}$ be the spatio-temporal samples taken at various time levels. The problem under consideration is to find necessary and sufficient conditions on $A, Omega, l_i$ in order to recover any $f in ell^2(I)$ from the measurements $Y$. This is the so called Dynamical Sampling Problem in which we seek to recover a signal $f$ by combining coarse samples of $f$ and its futures states $A^lf$. Various versions of dynamical sampling problems exhibit features that are similar to many fundamental problems: deconvolution, filter banks, super-resolution, compressed sensing etc. In this dissertation, we will study these problems.