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In this work, spin transport in the XXZ model is investigated. As a quantum integrable model the XXZ spin chain contains infinitely many conserved charges. The presence of these conserved charges results in the coexistence of ballistic and diffusive spin transport. Ballistic spin transport is characterized by the spin Drude weight, which was originally derived in 1999 by Zotos. This original derivation was controversial, particularly when divergences were observed in intermediate steps. In this thesis a novel approach for deriving the spin Drude weight demonstrates that Zotos' formula follows from relatively few assumptions. Furthermore, an asymptotic analysis of this formula at infinite temperature was found to be consistent with the Prosen--Ilievski bound, obtained from direct consideration of conserved charges. At low--temperatures a more complicated analysis is carried out to determine the non--analytic temperature correction term. This non--analytic temperature correction was found to give rise to the fractal structure of the Drude weight observed at finite temperatures. The form of this low--temperature correction raises important questions about the applicability of non--linear Luttinger liquid theory to quantum integrable models. Spin transport in the non--linear response framework is also considered in this thesis, where the non--linear Drude weight (NLDW) at infinite temperatures is determined. At infinite temperatures the NLDW is found to have an even more fractal structure than the linear Drude weight due to Bethe string lengths appearing explicitly in the final result. These results suggest that the Bethe strings may be indispensable for understanding spin transport.

The Nosé-Hoover thermostat is a widely-used technique in molecular-dynamics simulations for maintaining a canonical ensemble at a desired temperature. The associated differential equation, known as the Nosé-Hoover equation, plays a crucial role in understanding the dynamics of the system. This thesis investigates the integrability of the Nosé-Hoover equations using the Darboux theory of integrability. It presents original proofs of the Darboux non-integrability of these equations (thereby identifying and correcting several major mistakes in the previous work by Adam Mahdi and Claudia Valls. “Integrability of the Nosé–Hoover equation”. In: Journal of Geometry and Physics 61.8 (2011), pp. 1348–1352.).

The Nosé-Hoover thermostat is a widely-used technique in molecular-dynamics simulations for maintaining a canonical ensemble at a desired temperature. The associated differential equation, known as the Nosé-Hoover equation, plays a crucial role in understanding the dynamics of the system. This thesis investigates the integrability of the Nosé-Hoover equations using the Darboux theory of integrability. It presents original proofs of the Darboux non-integrability of these equations (thereby identifying and correcting several major mistakes in the previous work by Adam Mahdi and Claudia Valls. “Integrability of the Nosé–Hoover equation”. In: Journal of Geometry and Physics 61.8 (2011), pp. 1348–1352.).

In this work, spin transport in the XXZ model is investigated. As a quantum integrable model the XXZ spin chain contains infinitely many conserved charges. The presence of these conserved charges results in the coexistence of ballistic and diffusive spin transport. Ballistic spin transport is characterized by the spin Drude weight, which was originally derived in 1999 by Zotos. This original derivation was controversial, particularly when divergences were observed in intermediate steps. In this thesis a novel approach for deriving the spin Drude weight demonstrates that Zotos' formula follows from relatively few assumptions. Furthermore, an asymptotic analysis of this formula at infinite temperature was found to be consistent with the Prosen--Ilievski bound, obtained from direct consideration of conserved charges. At low--temperatures a more complicated analysis is carried out to determine the non--analytic temperature correction term. This non--analytic temperature correction was found to give rise to the fractal structure of the Drude weight observed at finite temperatures. The form of this low--temperature correction raises important questions about the applicability of non--linear Luttinger liquid theory to quantum integrable models. Spin transport in the non--linear response framework is also considered in this thesis, where the non--linear Drude weight (NLDW) at infinite temperatures is determined. At infinite temperatures the NLDW is found to have an even more fractal structure than the linear Drude weight due to Bethe string lengths appearing explicitly in the final result. These results suggest that the Bethe strings may be indispensable for understanding spin transport.

Stephen Kudla has conjectured a relationship between the Fourier coefficients of Eisenstein series, and the arithmetic heights of certain special cycles. Luis Garcia and Siddarth Sankaran confirmed the conjecture for certain Shimura varieties of type U(p, q), arising as the quotient of a symmetric space by a group action, when q=1. An essential step in their argument relies on establishing that a specific form is a highest weight vector of a particular weight, for the Weil representation. In an effort to extend the results of Garcia and Sankaran, we show that the aforementioned forms are highest weight vectors of the expected weight, under the action of the Weil representation, in various cases when q>1. In particular, we show that this result holds for all cases when q=2. We prove this result by using an inductive argument, which depends on a technical result about immersed submanifolds, and various results about splitting the action of the Weil representation on tensor products. The base cases are intractable to carry out by hand, and thus the final section of the thesis contains Sage code which was written to carry out the computations of the base cases.

Stephen Kudla has conjectured a relationship between the Fourier coefficients of Eisenstein series, and the arithmetic heights of certain special cycles. Luis Garcia and Siddarth Sankaran confirmed the conjecture for certain Shimura varieties of type U(p, q), arising as the quotient of a symmetric space by a group action, when q=1. An essential step in their argument relies on establishing that a specific form is a highest weight vector of a particular weight, for the Weil representation. In an effort to extend the results of Garcia and Sankaran, we show that the aforementioned forms are highest weight vectors of the expected weight, under the action of the Weil representation, in various cases when q>1. In particular, we show that this result holds for all cases when q=2. We prove this result by using an inductive argument, which depends on a technical result about immersed submanifolds, and various results about splitting the action of the Weil representation on tensor products. The base cases are intractable to carry out by hand, and thus the final section of the thesis contains Sage code which was written to carry out the computations of the base cases.

Let Sigma be a bordered Riemann surface of genus g > 0 with n>1 borders Gamma_1, ..., Gamma_n each one homeomorphic to the unit circle. This surface can be described as a compact Riemann surface R of the same genus with finitely many simply connected domains Omega_1, ..., Omega_n removed. Let f_k be a conformal map from the unit disc D in C onto Omega_k, k =1,..., n. We first generalize the classical Faber and Grunsky operators, to operators on Sigma associated to the maps f = (f_1, ..., f_n). These two operators are used to characterize the holomorphic Dirichlet space D(Sigma). More precisely, we show that the pull-back of the conformally-non-tangential boundary values of functions in D(Sigma) under f is the graph of the Grunsky operator. We also show that the Grunsky operator is a bounded operator of norm strictly less than one. So far this had only be proven for the genus zero case. The central problem is to prove that the Faber operator is a bounded isomorphism. This is done by using the Schiffer operators, which we generalize it for Sigma in this work. We characterize the function space on which the Schiffer operator is a bounded isomorphism. This characterization depends on the topology of the surface. The condition that the boundary curves are quasicircles plays a vital role in this proof. This is also a generalization of the genus zero case. The Grunsky operator is used to define a map, say Pi_g, on the Teichmüller space of Sigma. The map Pi_g has some analogies with the classical period map defined for compact surfaces. We conclude the thesis with a conjecture regarding the holomorphicity of this map for g, which was an important source of motivation for the results of the thesis.

The Standard Model (SM) is currently our most complete, fundamental, and successful theory of nature. However, despite its ability to make some of the most precise predictions in all the physical sciences, there are scenarios where it has been shown to be inadequate or incomplete. This is why SM Precision tests are very important avenues to a solution because they can inform us where the SM is incorrect, and in a manner of speaking, by how much. In this dissertation, we look at two major precision tests: the weak mixing angle and the unitarity condition sum of the top row CKM matrix elements. In order to extract either of these quantities at the level of precision required to meaningfully test the SM, we must go beyond their leading order calculation, and compute all of their 1 loop radiative corrections. The physical process we consider acts like a host to perform both the calculation and the experiment and must be somewhat practical. Semileptonic reactions are a prime candidate as nucleons and nuclei provide practical targets in scattering experiments as well as ideal decay parents, as opposed to their leptonic alternatives. The cost of this experimental convenience are additional challenges when calculating the observables in the SM theory, as hadronic modeling becomes a necessity. This dissertation investigates some of the most troublesome radiative corrections in an attempt to reduce their hadronic uncertainties, using state-of-the-art dispersive techniques as well as update some previously calculated radiative corrections in the aforementioned test quantities. Using these methods, we have found that the CKM matrix is no longer consistent with unitarity and an updated relationship between the weak mixing angle and the weak charge of the proton is proposed.

This thesis provides a review of left/right orderable and circularly orderable groups. A detailed proof of the classical Farrell’s theorem for left orderable groups is given, and is generalized to the case of circularly orderable groups through a cohomological argument. This generalizes the classical Farrell’s theorem, in the sense that left orderable group may be viewed as the special case of circularly orderable group.

The Standard Model (SM) is currently our most complete, fundamental, and successful theory of nature. However, despite its ability to make some of the most precise predictions in all the physical sciences, there are scenarios where it has been shown to be inadequate or incomplete. This is why SM Precision tests are very important avenues to a solution because they can inform us where the SM is incorrect, and in a manner of speaking, by how much. In this dissertation, we look at two major precision tests: the weak mixing angle and the unitarity condition sum of the top row CKM matrix elements. In order to extract either of these quantities at the level of precision required to meaningfully test the SM, we must go beyond their leading order calculation, and compute all of their 1 loop radiative corrections. The physical process we consider acts like a host to perform both the calculation and the experiment and must be somewhat practical. Semileptonic reactions are a prime candidate as nucleons and nuclei provide practical targets in scattering experiments as well as ideal decay parents, as opposed to their leptonic alternatives. The cost of this experimental convenience are additional challenges when calculating the observables in the SM theory, as hadronic modeling becomes a necessity. This dissertation investigates some of the most troublesome radiative corrections in an attempt to reduce their hadronic uncertainties, using state-of-the-art dispersive techniques as well as update some previously calculated radiative corrections in the aforementioned test quantities. Using these methods, we have found that the CKM matrix is no longer consistent with unitarity and an updated relationship between the weak mixing angle and the weak charge of the proton is proposed.

The Standard Model (SM) is currently our most complete, fundamental, and successful theory of nature. However, despite its ability to make some of the most precise predictions in all the physical sciences, there are scenarios where it has been shown to be inadequate or incomplete. This is why SM Precision tests are very important avenues to a solution because they can inform us where the SM is incorrect, and in a manner of speaking, by how much. In this dissertation, we look at two major precision tests: the weak mixing angle and the unitarity condition sum of the top row CKM matrix elements. In order to extract either of these quantities at the level of precision required to meaningfully test the SM, we must go beyond their leading order calculation, and compute all of their 1 loop radiative corrections. The physical process we consider acts like a host to perform both the calculation and the experiment and must be somewhat practical. Semileptonic reactions are a prime candidate as nucleons and nuclei provide practical targets in scattering experiments as well as ideal decay parents, as opposed to their leptonic alternatives. The cost of this experimental convenience are additional challenges when calculating the observables in the SM theory, as hadronic modeling becomes a necessity. This dissertation investigates some of the most troublesome radiative corrections in an attempt to reduce their hadronic uncertainties, using state-of-the-art dispersive techniques as well as update some previously calculated radiative corrections in the aforementioned test quantities. Using these methods, we have found that the CKM matrix is no longer consistent with unitarity and an updated relationship between the weak mixing angle and the weak charge of the proton is proposed.

Let Sigma be a bordered Riemann surface of genus g > 0 with n>1 borders Gamma_1, ..., Gamma_n each one homeomorphic to the unit circle. This surface can be described as a compact Riemann surface R of the same genus with finitely many simply connected domains Omega_1, ..., Omega_n removed. Let f_k be a conformal map from the unit disc D in C onto Omega_k, k =1,..., n. We first generalize the classical Faber and Grunsky operators, to operators on Sigma associated to the maps f = (f_1, ..., f_n). These two operators are used to characterize the holomorphic Dirichlet space D(Sigma). More precisely, we show that the pull-back of the conformally-non-tangential boundary values of functions in D(Sigma) under f is the graph of the Grunsky operator. We also show that the Grunsky operator is a bounded operator of norm strictly less than one. So far this had only be proven for the genus zero case. The central problem is to prove that the Faber operator is a bounded isomorphism. This is done by using the Schiffer operators, which we generalize it for Sigma in this work. We characterize the function space on which the Schiffer operator is a bounded isomorphism. This characterization depends on the topology of the surface. The condition that the boundary curves are quasicircles plays a vital role in this proof. This is also a generalization of the genus zero case. The Grunsky operator is used to define a map, say Pi_g, on the Teichmüller space of Sigma. The map Pi_g has some analogies with the classical period map defined for compact surfaces. We conclude the thesis with a conjecture regarding the holomorphicity of this map for g, which was an important source of motivation for the results of the thesis.

This thesis provides a review of left/right orderable and circularly orderable groups. A detailed proof of the classical Farrell’s theorem for left orderable groups is given, and is generalized to the case of circularly orderable groups through a cohomological argument. This generalizes the classical Farrell’s theorem, in the sense that left orderable group may be viewed as the special case of circularly orderable group.

The Standard Model (SM) is currently our most complete, fundamental, and successful theory of nature. However, despite its ability to make some of the most precise predictions in all the physical sciences, there are scenarios where it has been shown to be inadequate or incomplete. This is why SM Precision tests are very important avenues to a solution because they can inform us where the SM is incorrect, and in a manner of speaking, by how much. In this dissertation, we look at two major precision tests: the weak mixing angle and the unitarity condition sum of the top row CKM matrix elements. In order to extract either of these quantities at the level of precision required to meaningfully test the SM, we must go beyond their leading order calculation, and compute all of their 1 loop radiative corrections. The physical process we consider acts like a host to perform both the calculation and the experiment and must be somewhat practical. Semileptonic reactions are a prime candidate as nucleons and nuclei provide practical targets in scattering experiments as well as ideal decay parents, as opposed to their leptonic alternatives. The cost of this experimental convenience are additional challenges when calculating the observables in the SM theory, as hadronic modeling becomes a necessity. This dissertation investigates some of the most troublesome radiative corrections in an attempt to reduce their hadronic uncertainties, using state-of-the-art dispersive techniques as well as update some previously calculated radiative corrections in the aforementioned test quantities. Using these methods, we have found that the CKM matrix is no longer consistent with unitarity and an updated relationship between the weak mixing angle and the weak charge of the proton is proposed.

In this thesis, we investigate the period mapping of Teichmuller space into the Siegel upper half space. This is constructed from integrals of a basis of holomorphic one-forms along closed curves of a basis of the Riemann surface. We consider the Riemann, Teichmuller and Torelli moduli spaces and their representation in the Siegel upper half space, and its relation to orbits of a symplectic and a set of positive polarizations of a vector space of dimension equal to the genus of the surface.

In this thesis, we investigate the period mapping of Teichmuller space into the Siegel upper half space. This is constructed from integrals of a basis of holomorphic one-forms along closed curves of a basis of the Riemann surface. We consider the Riemann, Teichmuller and Torelli moduli spaces and their representation in the Siegel upper half space, and its relation to orbits of a symplectic and a set of positive polarizations of a vector space of dimension equal to the genus of the surface.

In the present work we develop a unified way of looking at Normed Spaces of Continuous and Holomorphic Functions (NSCF’s and NSHF’s) and of Weighted Composition Operators (WCO’s) between these spaces. If F is a normed space of continuous functions over a topological space X, we relate different properties of F. We show that compactness of the inclusion map JF from F into C(X) is equivalent to continuity of the operation of evaluation on BF×X, where BF is the closed unit ball of F endowed with the weak topology. Moreover, we show that these conditions follow from mere continuity of the norm of the point evaluations, under the additional assumption that F is a uniformly smooth Banach space (this was previously known for Hilbert spaces). Also, we show that F is reflexive if and only if JF is weakly compact and BF is closed in C(X) in any equivalent norm on F. On the other hand, we show that some mild conditions on F imply that X is metrizable. We also give some sufficient conditions that force WCO’s between NSCF’s to have continuous symbols (this problem was previously considered only for a specific choice of NSCF’s). We provide counterexamples that show that the problem is not trivial. An analogous problem is considered in the holomorphic setting. We also show that on a wide class of NSCF’s the only unitary multiplication operators are the multiples of identity. For a Reproducing Kernel Hilbert Space (RKHS) of holomorphic functions on a complex domain we give a formula that describes the Hermitean metrics on the domain which are pullbacks of some metric on the (dual of) the RKHS via the evaluation map. Then we consider the question when such metrics are invariant with respect to the group of automorphisms of the domain, and obtain some partial results in that direction.

In the present work we develop a unified way of looking at Normed Spaces of Continuous and Holomorphic Functions (NSCF’s and NSHF’s) and of Weighted Composition Operators (WCO’s) between these spaces. If F is a normed space of continuous functions over a topological space X, we relate different properties of F. We show that compactness of the inclusion map JF from F into C(X) is equivalent to continuity of the operation of evaluation on BF×X, where BF is the closed unit ball of F endowed with the weak topology. Moreover, we show that these conditions follow from mere continuity of the norm of the point evaluations, under the additional assumption that F is a uniformly smooth Banach space (this was previously known for Hilbert spaces). Also, we show that F is reflexive if and only if JF is weakly compact and BF is closed in C(X) in any equivalent norm on F. On the other hand, we show that some mild conditions on F imply that X is metrizable. We also give some sufficient conditions that force WCO’s between NSCF’s to have continuous symbols (this problem was previously considered only for a specific choice of NSCF’s). We provide counterexamples that show that the problem is not trivial. An analogous problem is considered in the holomorphic setting. We also show that on a wide class of NSCF’s the only unitary multiplication operators are the multiples of identity. For a Reproducing Kernel Hilbert Space (RKHS) of holomorphic functions on a complex domain we give a formula that describes the Hermitean metrics on the domain which are pullbacks of some metric on the (dual of) the RKHS via the evaluation map. Then we consider the question when such metrics are invariant with respect to the group of automorphisms of the domain, and obtain some partial results in that direction.

We define a Yang-Mills model in 1+1 dimensions coupled to a real scalar field and we study the Yang-Mills flow equations for this simple model. Yang-Mills flows have not been thoroughly studied, especially in a physical context, but may be able to provide valuable insight into both particle physics as well as gravity. We study our model using both the Hamiltonian equations and Euler-Lagrange equations, and we calculate the flow numerically using a simple finite difference method for the case of an Abelian Lie group and static fields. We are able to find several analytic solutions to the equations of motion and the numerical calculation of the flow suggests most non-constant solutions are unstable. We also find that the flow depends upon the relative values of the coupling constant and the mass of the scalar field. The results found with this simple model provide a starting point for the study of Yang-Mills flow in the context of more complicated (but more physical) models such as the Abelian Higgs.

The Qweak experiment will provide the most precise determination of the proton’s weak charge Q^p_W by measuring the parity-violating asymmetry in elastic electron-proton scattering at low momentum transfer Q^2 = 0.0250 (GeV/c)^2. Qpw is related to the weak mixing angle sin^2\theta_W, a fundamental parameter of the standard model. A final measurement of the weak charge at the proposed 4% relative uncertainty is sensitive to certain types of new parity-violating physics beyond the standard model at the TeV mass-scale. Data were taken over a two year period beginning in 2010, using a custom apparatus installed in Hall-C at the Thomas Jefferson National Accelerator Facility in Newport News, Virginia. A 180 microamp beam of longitudinally polarized electrons was accelerated to 1.165 GeV and then scattered from unpolarized protons in a liquid hydrogen target. The scattered electrons were then collimated and steered using a magnetic spectrometer onto a set of azimuthally symmetric quartz bar Cherenkov detectors. The performance of this main detector subsystem will be described in detail in this dissertation. A blinded analysis of Run-II, roughly 2/3 of the entire Qweak data set, resulted in an elastic ep asymmetry of -235.6 +/- 8.7 (Stat) +/- 9.3 (Syst.) +/- 39.3 (Blind) ppb. Using this value, the proton’s weak charge was calculated to be QpW = 0.0714 +/- 0.0093. This constitutes a 17% relative measurement, that will reduce to < 9% upon unblinding the asymmetry. This dissertation will provide the details that went into extracting both the parity-violating asymmetry and the weak charge of the proton.

We define a Yang-Mills model in 1+1 dimensions coupled to a real scalar field and we study the Yang-Mills flow equations for this simple model. Yang-Mills flows have not been thoroughly studied, especially in a physical context, but may be able to provide valuable insight into both particle physics as well as gravity. We study our model using both the Hamiltonian equations and Euler-Lagrange equations, and we calculate the flow numerically using a simple finite difference method for the case of an Abelian Lie group and static fields. We are able to find several analytic solutions to the equations of motion and the numerical calculation of the flow suggests most non-constant solutions are unstable. We also find that the flow depends upon the relative values of the coupling constant and the mass of the scalar field. The results found with this simple model provide a starting point for the study of Yang-Mills flow in the context of more complicated (but more physical) models such as the Abelian Higgs.

The Qweak experiment will provide the most precise determination of the proton’s weak charge Q^p_W by measuring the parity-violating asymmetry in elastic electron-proton scattering at low momentum transfer Q^2 = 0.0250 (GeV/c)^2. Qpw is related to the weak mixing angle sin^2\theta_W, a fundamental parameter of the standard model. A final measurement of the weak charge at the proposed 4% relative uncertainty is sensitive to certain types of new parity-violating physics beyond the standard model at the TeV mass-scale. Data were taken over a two year period beginning in 2010, using a custom apparatus installed in Hall-C at the Thomas Jefferson National Accelerator Facility in Newport News, Virginia. A 180 microamp beam of longitudinally polarized electrons was accelerated to 1.165 GeV and then scattered from unpolarized protons in a liquid hydrogen target. The scattered electrons were then collimated and steered using a magnetic spectrometer onto a set of azimuthally symmetric quartz bar Cherenkov detectors. The performance of this main detector subsystem will be described in detail in this dissertation. A blinded analysis of Run-II, roughly 2/3 of the entire Qweak data set, resulted in an elastic ep asymmetry of -235.6 +/- 8.7 (Stat) +/- 9.3 (Syst.) +/- 39.3 (Blind) ppb. Using this value, the proton’s weak charge was calculated to be QpW = 0.0714 +/- 0.0093. This constitutes a 17% relative measurement, that will reduce to < 9% upon unblinding the asymmetry. This dissertation will provide the details that went into extracting both the parity-violating asymmetry and the weak charge of the proton.

In this thesis we will consider and investigate the properties of analytic functions via their behavior near the boundary of the domain on which they are defined. To do that we introduce the notion of the hyperbolic distortion and the hyperbolic derivative. Classical results state that the hyperbolic derivative is bounded from above by 1, and we will consider the case when it is bounded from below by some positive constant. Boundedness from below of the hyperbolic derivative implies some nice properties of the function near the boundary. For instance Krauss & all in 2007 proved that, if the function is defined on a domain bounded by analytic curve, then boundedness from below of the hyperbolic derivative implies that the function has an analytic continuation across the boundary. We extend this result for the domains with slightly more general boundary, namely for smooth Jordan domains, and get that in this case the function and its derivative will have only continuous extensions to the boundary.

Some branches of quantum gravity demand the existence of higher dimensions and the addition of higher curvature terms to the gravitational Lagrangian in the form of the Lovelock polynomials. In this thesis we investigate some of the classical properties of Lovelock gravity. We first derive the Hamiltonian for Lovelock gravity and find that it takes the same form as in general relativity when written in terms of the Misner-Sharp mass function. We then minimally couple the action to matter fields to find Hamilton’s equations of motion. These are gauge fixed to be in the Painleve-Gullstrand co–ordinates and are well suited to numerical studies of black hole formation. We then use these equations of motion for the massless scalar field to study the formation of general relativistic black holes in four to eight dimensions and Einstein-Gauss-Bonnet black holes in five and six dimensions. We study Choptuik scaling, a phenomenon which relates the initial conditions of a matter distribution to the final observables of small black holes. In both higher dimensional general relativity and Einstein-Gauss-Bonnet gravity we confirm the existence of cusps in the mass scaling relation which had previously only been observed in four dimensional general relativity. In the general relativistic case we then calculate the critical exponents for four to eight dimensions and find agreement with previous calculations by Bland et al but not Sorkin et al who both worked in null co–ordinates. For the Einstein-Gauss-Bonnet case we find that the self-similar behaviour seen in the general relativistic case is destroyed. We find that it is replaced by some other form of scaling structure. In five dimensions we find that the period of the critical solution at the origin is proportional to roughly the cube root of the Gauss-Bonnet parameter and that there is evidence for a minimum black hole radius. In six dimensions we see evidence for a new type of scaling. We also show, from the equations of moti

In this thesis we will consider and investigate the properties of analytic functions via their behavior near the boundary of the domain on which they are defined. To do that we introduce the notion of the hyperbolic distortion and the hyperbolic derivative. Classical results state that the hyperbolic derivative is bounded from above by 1, and we will consider the case when it is bounded from below by some positive constant. Boundedness from below of the hyperbolic derivative implies some nice properties of the function near the boundary. For instance Krauss & all in 2007 proved that, if the function is defined on a domain bounded by analytic curve, then boundedness from below of the hyperbolic derivative implies that the function has an analytic continuation across the boundary. We extend this result for the domains with slightly more general boundary, namely for smooth Jordan domains, and get that in this case the function and its derivative will have only continuous extensions to the boundary.

Some branches of quantum gravity demand the existence of higher dimensions and the addition of higher curvature terms to the gravitational Lagrangian in the form of the Lovelock polynomials. In this thesis we investigate some of the classical properties of Lovelock gravity. We first derive the Hamiltonian for Lovelock gravity and find that it takes the same form as in general relativity when written in terms of the Misner-Sharp mass function. We then minimally couple the action to matter fields to find Hamilton’s equations of motion. These are gauge fixed to be in the Painleve-Gullstrand co–ordinates and are well suited to numerical studies of black hole formation. We then use these equations of motion for the massless scalar field to study the formation of general relativistic black holes in four to eight dimensions and Einstein-Gauss-Bonnet black holes in five and six dimensions. We study Choptuik scaling, a phenomenon which relates the initial conditions of a matter distribution to the final observables of small black holes. In both higher dimensional general relativity and Einstein-Gauss-Bonnet gravity we confirm the existence of cusps in the mass scaling relation which had previously only been observed in four dimensional general relativity. In the general relativistic case we then calculate the critical exponents for four to eight dimensions and find agreement with previous calculations by Bland et al but not Sorkin et al who both worked in null co–ordinates. For the Einstein-Gauss-Bonnet case we find that the self-similar behaviour seen in the general relativistic case is destroyed. We find that it is replaced by some other form of scaling structure. In five dimensions we find that the period of the critical solution at the origin is proportional to roughly the cube root of the Gauss-Bonnet parameter and that there is evidence for a minimum black hole radius. In six dimensions we see evidence for a new type of scaling. We also show, from the equations of moti

In this thesis we will consider and investigate the properties of analytic functions via their behavior near the boundary of the domain on which they are defined. To do that we introduce the notion of the hyperbolic distortion and the hyperbolic derivative. Classical results state that the hyperbolic derivative is bounded from above by 1, and we will consider the case when it is bounded from below by some positive constant. Boundedness from below of the hyperbolic derivative implies some nice properties of the function near the boundary. For instance Krauss & all in 2007 proved that, if the function is defined on a domain bounded by analytic curve, then boundedness from below of the hyperbolic derivative implies that the function has an analytic continuation across the boundary. We extend this result for the domains with slightly more general boundary, namely for smooth Jordan domains, and get that in this case the function and its derivative will have only continuous extensions to the boundary.

In this thesis we will consider and investigate the properties of analytic functions via their behavior near the boundary of the domain on which they are defined. To do that we introduce the notion of the hyperbolic distortion and the hyperbolic derivative. Classical results state that the hyperbolic derivative is bounded from above by 1, and we will consider the case when it is bounded from below by some positive constant. Boundedness from below of the hyperbolic derivative implies some nice properties of the function near the boundary. For instance Krauss & all in 2007 proved that, if the function is defined on a domain bounded by analytic curve, then boundedness from below of the hyperbolic derivative implies that the function has an analytic continuation across the boundary. We extend this result for the domains with slightly more general boundary, namely for smooth Jordan domains, and get that in this case the function and its derivative will have only continuous extensions to the boundary.

Let f be an analytic and one-to-one function on the unit disk such that f(0)=0. Let Q(w)dw^2 be a quadratic differential. Suppose that f maps into the complex plane or the unit disk minus analytic arcs w(t) satisfying Q(w(t))(dw/dt)^2<0. We are interested in the question: if Q is unknown but of a specified form, does f determine the quadratic differential Q uniquely? Our main result is that for functions mapping into the unit disk and quadratic differentials with a pole of order 4 at the origin, the quadratic differential is uniquely determined up to exceptional cases. This question arises in the study of extremal functions for functionals over classes of analytic one-to-one maps.

Let f be an analytic and one-to-one function on the unit disk such that f(0)=0. Let Q(w)dw^2 be a quadratic differential. Suppose that f maps into the complex plane or the unit disk minus analytic arcs w(t) satisfying Q(w(t))(dw/dt)^2<0. We are interested in the question: if Q is unknown but of a specified form, does f determine the quadratic differential Q uniquely? Our main result is that for functions mapping into the unit disk and quadratic differentials with a pole of order 4 at the origin, the quadratic differential is uniquely determined up to exceptional cases. This question arises in the study of extremal functions for functionals over classes of analytic one-to-one maps.

The imaginary part of the retarded self energy is of particular interest as it contains a lot of physical information about particle interactions. In higher order loop diagrams the calculation become extremely tedious and if we have to do the same at finite temperature, it includes an extra dimension to the difficulty. In such a condition we require to switch between bases and select the best basis for a particular diagram. We have shown in our calculation that in higher order loop diagrams, at finite temperature, the R/A basis is most convenient on summing over the internal vertices and very efficient on calculating some particular diagrams while the result is most easily interpretable in the Keldysh basis for most other complex diagrams.

The sewing operation is an integral component of both Geometric Function Theory and Conformal Field Theory and in this thesis we explore the interplay between the two fields. We will first generalize Huang's Geometric Sewing Equation to the quasi-symmetric case. That is, given specific maps g(z) and f^{-1}(z), we show the existence of the sewing maps F_{1}(z) and F_{2}(z). Second, we display an algebraic procedure using convergent matrix operations showing that the coefficients of the Conformal Welding Theorem maps F(z) and G(z) are dependent on the coefficients of the map phi(z). We do this for both the analytic and quasi-symmetric cases, and it is done using a special block/vector decomposition of a matrix representation called the power matrix. Lastly, we provide a partial result: given specific maps g(z) and f^{-1}(z) with analytic extensions, as well as a particular analytic map phi(z), it is possible to provide a method of determining the coefficients of the complementary maps.

The imaginary part of the retarded self energy is of particular interest as it contains a lot of physical information about particle interactions. In higher order loop diagrams the calculation become extremely tedious and if we have to do the same at finite temperature, it includes an extra dimension to the difficulty. In such a condition we require to switch between bases and select the best basis for a particular diagram. We have shown in our calculation that in higher order loop diagrams, at finite temperature, the R/A basis is most convenient on summing over the internal vertices and very efficient on calculating some particular diagrams while the result is most easily interpretable in the Keldysh basis for most other complex diagrams.