MSc., Faculty of Science, University of the Witwatersrand, 2011 / We try to develop a collective field theory of single matrix models by using the formalism of Jevicki and Sakita in [1], with Schur polynomials as our collective fields. Field operators and the relation for the change of variables required to obtain the collective field Hamiltonian are found using group representation theory.

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Schramm-Loewner evolution (SLE(kappa)) is an important contemporary tool for identifying critical scaling limits of two-dimensional statistical systems. The SLE(kappa) one-parameter family of processes can be viewed as a special case of a more general, two-parameter family of processes we denote SLE(kappa, mu). The SLE(kappa, mu) process is defined for kappa>0 and real numbers mu; it represents the solution of the chordal Loewner equations under special conditions on the driving function parameter which require that it is a Brownian motion with drift mu and variance kappa. We derive properties of this process by use of methods applied to SLE(kappa) and application of Girsanov's Theorem. In contrast to SLE(kappa), we identify stationary asymptotic behavior of SLE(kappa, mu). For kappa in (0,4] and mu > 0, we present a pathwise construction of a process with stationary temporal increments and stationary imaginary component and relate it to the limiting behavior of the SLE(kappa, mu) generating curve. Our main result is a spatial invariance property of this process achieved by defining a top-crossing probability for points in the upper half plane with respect to the generating curve.

A thesis submitted to the Faculty of Science, University of the Witwatersrand, Johannesburg, in fulfilment of the requirements for the degree of Doctor of Philosophy. Johannesburg, 2016. / The convenient pictorial descriptions of the half-BPS and near-BPS sectors of the AdS=CFT equivalent theories of N = 4, D = 4 super Yang-Mills and D = 10 Type IIB superstring theory on AdS5 S5 are exploited in this thesis by using Schur polynomials labelled by Young diagrams as a basis for the gauge invariant operators in the eld theory. We use a \Fourier transform" on these operators to construct asymptotic eigenstates of the dilatation operator, the spectrum of which agrees precisely with the rst two leading order terms in the smallcoupling expansion of the exact result determined by symmetry. Motivated by the geometric description of the systems of open strings with magnon excitations to which the operators are dual, we propose a simple and minimal all-loop expression that interpolates between anomalous dimensions computed in the gauge theory and energies computed in the string theory. The connection to the string theory result provides the insight necessary to understand the interpretation of our Gauss graphs in the magnon language. Symmetry determines the two-body scattering matrix for the magnons up to a phase, and it is demonstrated that integrability is spoiled by the boundary conditions on the open strings. The Schur polynomial construction is then applied to the study of closed strings on a class of half- BPS excitations of the AdS5 S5 background. The string theory predictions for the magnon energies are again reproduced by calculating the anomalous dimensions of particular linear combinations of our operators. Group theoretic quantities which can be read o the Young diagram labels provide the correct modi cation of terms in the dilatation action to account for the energies of magnons at di erent radii on the LLM plane. The representation theory implies a natural splitting of the full symmetry group - the distinction between what is the background and what is the excitation is accomplished in the choice of the subgroup and representations used to construct the operator. Connecting the descriptions utilised in obtaining these results is expected to allow the construction of operators dual to general open string con gurations on the class of backgrounds considered. / GR 2016

Core-collapse supernovae (CCSN) are among the most energetic explosions in the universe, liberating ~1053 erg of gravitational binding energy of the stellar core. Most of this energy ( ~99%) is emitted in neutrinos and only 1% is released as electromagnetic radiation in the visible spectrum. Energy radiated in the form of gravitational waves (GWs) is about five orders smaller. Nevertheless, this energy corresponds to a very strong GW signal and, because of this CCSN are considered as one of the prime sources of gravitational waves for interferometric detectors. Gravitational waves can give us access to the electromagnetically hidden compact inner core of supernovae. They will provide valuable information about the angular momentum distribution and the baryonic equation of state, both of which are uncertain. Furthermore, they might even help to constrain theoretically predicted SN mechanisms. Detection of GW signals and analysis of the observations will require realistic signal predi ctions from the non-parameterized relativistic numerical simulations of CCSN. This dissertation presents the gravitational wave signature of core-collapse v supernovae. Previous studies have considered either parametric models or nonexploding models of CCSN. This work presents complete waveforms, through the explosion phase, based on first-principles models for the first time. We performed 2D simulations of CCSN using the CHIMERA code for 12, 15, and 25M non-rotating progenitors. CHIMERA incorporates most of the criteria for realistic core-collapse modeling, such as multi-frequency neutrino transport coupled with relativistic hydrodynamics, eective GR potential, nuclear reaction network, and an industry-standard equation of state. / Based on the results of our simulations, I produced the most realistic gravitational waveforms including all postbounce phases of core-collapse supernovae: the prompt convection, the stationary accretion shock instability, and the corresponding explosion. Additionally, the tracer particles applied in the analysis of the GW signal reveal the origin of low-frequency component in the prompt part of gravitational waveform. Analysis of detectability of the GW signature from a Galactic event shows that the signal is within the band-pass of current and future GW observatories such as AdvLIGO, advanced Virgo, and LCGT. / by Konstantin Yakunin. / Thesis (Ph.D.)--Florida Atlantic University, 2011. / Includes bibliography. / Electronic reproduction. Boca Raton, Fla., 200?. Mode of access: World Wide Web.

Orientador: José Francisco Gomes / Banca: Abraham Hirsz Zimerman / Banca: Eliezer Batista / Resumo: Nesta dissertação estudamos as estruturas geométricas e algébricas subjacentes aos modelos de Toda. Primeiramente, vemos como as equações de Toda são consequência da condição de curvatura nula de um certo fibrado principal holomórfico e posteriormente, introduzimos a formulação Lagrangiana dos mesmos, como perturbações integráveis de um modelo de WZW calibrado num espaço quociente. Terminamos com um estudo da dualidade própria destas teorias / Abstract: In this work we study the differential geometry formulation of Toda models. Firstly showing how the Toda equations are consequence of the zero curvature condition of a given holomorfic principal bundle and later introducing the Lagrangian formulation of the Toda models as integrable perturbations of a gauged WZW model in a special coset. We end up with a study of the duality properties of such class of theories / Mestre

Orientador: Maria Cristina Batoni Abdalla Ribeiro / Coorientador: Andrey Alexandrovich Bytsenko / Banca: Ruben Aldrovandi / Banca: José Abdalla Helayel-Neto / Resumo: Apresentamos nesta dissertação uma revisão dos conceitos de geometria diferencial, onde estamos interessados em definir campos vetoriais que geram transformações de um parâmetro, formas diferenciais, variedades simpléticas e fibrados. Além disso, detalhamos o conceito de cohomologia de De Rham, o qual nos fornece uma ferramenta algébrica fundamental para analisar propriedades topológicas das variedades. A combinação desses conceitos, os quais suportam o nosso trabalho, permite-nos desenvolver teorias de localização equivariante de integrais definidas sobre espaços de fase clássicos, os quais também podem ser uma órbita co-adjunta. A localização é possível devido ao teorema de Duistermaat-Heckman, o qual nos permite escrever integrais como uma soma, ou integral, sobre o conjunto dos pontos críticos do espaço. Em seguida fazemos uma extensão para teorias de localização de integrais funcionais, onde é preciso definir o espaço dos loops. Nesse contexto aplicamos a formulação de localização equivariante tendo como base a conjectura de Atiyah-Witten para teorias supersimétricas, onde derivamos o teorema de índice de Atiyah-Singer para um operador de Dirac. O teorema de índice é aplicado no cálculo da anomalia quiral / Abstract: We present in this dissertation a conceptual review of differential geometry, where we are interested in defining vector fields which are one-parameter transformation generators, differential forms, symplectic manifolds, and fiber bundles. In addition, we detail the concept about De Rham's cohomology, which provides us a fundamental algebraic tool to analyze topological properties of manifolds. The combination of these concepts, which are the background material of our work, allows us to develop equivariant localization theories of integrals defined on classical phase spaces, which can also be a co-adjoint orbit. The localization is possible because of the Duistermaat-Heckman theorem, which allows us to write integrals on the whole space just as a sum, or integral, on a critical points set. Further more, we do an extension to functional integrals localization theories, where it is needed to define loop spaces. In this context we apply equivariant localization formulation having the bases of Atiyah-Witten conjecture to supersymmetric theories, where we derive the Atiyah-Singer index theorem for a Dirac operator. The index theorem is applied to chiral anomaly calculation / Mestre

Despite the high-dimensional nature of the nervous system, humans produce low-dimensional cognitive and behavioral dynamics. How high-dimensional networks with complex connectivity give rise to functionally meaningful dynamics is not well understood. How does a neural network encode function? How can functional dynamics be systematically obtained from networks? There exist few frameworks in the current literature that answer these questions satisfactorily. In this dissertation I propose a general theoretical framework entitled 'Structured Flows on Manifolds' and its underlying mathematical basis. The framework is based on the principles of non-linear dynamical systems and Synergetics and can be used to understand how high-dimensional systems that exhibit multiple time-scale behavior can produce low-dimensional dynamics. Low-dimensional functional dynamics arises as a result of the timescale separation of the systems component's dynamics. The low-dimensional space in which the functi onal dynamics occurs is regarded as a manifold onto which the entire systems dynamics collapses. For the duration of the function the system will stay on the manifold and evolve along the manifold. From a network perspective the manifold is viewed as the product of the interactions of the network nodes. The subsequent flows on the manifold are a result of the asymmetries of network's interactions. A distributed functional architecture based on this perspective is presented. Within this distributed functional architecture, issues related to networks such as flexibility, redundancy and robustness of the network's dynamics are addressed. Flexibility in networks is demonstrated by showing how the same network can produce different types of dynamics as a function of the asymmetrical coupling between nodes. Redundancy can be achieved by systematically creating different networks that exhibit the same dynamics. The framework is also used to systematically probe the effects of lesion / (removal of nodes) on network dynamics. It is also shown how low-dimensional functional dynamics can be obtained from firing-rate neuron models by placing biologically realistic constraints on the coupling. Finally the theoretical framework is applied to real data. Using the structured flows on manifolds approach we quantify team performance and team coordination and develop objective measures of team performance based on skill level. / by Ajay S. Pillai. / Thesis (Ph.D.)--Florida Atlantic University, 2008. / Includes bibliography. / Electronic reproduction. Boca Raton, FL : 2008 Mode of access: World Wide Web.

Lee Kwok San = 二維海森堡模型的熱力學性質之數値硏究 / 李國姗. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2001. / Includes bibliographical references (leaves 106-108). / Text in English; abstracts in English and Chinese. / Lee Kwok San = Er wei hai sen bao mo xing de re li xue xing zhi zhi shu zhi yan jiu / Li Guoshan. / Abstract --- p.i / Acknowledgement --- p.iii / Chapter 1 --- Introduction --- p.1 / Chapter 1.1 --- History of magnetism --- p.1 / Chapter 1.2 --- History of Heisenberg model --- p.2 / Chapter 1.3 --- Heisenberg model and high-Tc superconductors --- p.6 / Chapter 1.4 --- Organization of thesis --- p.8 / Chapter 2 --- Methodology --- p.10 / Chapter 2.1 --- Introduction --- p.10 / Chapter 2.2 --- Exact diagonalization --- p.11 / Chapter 2.2.1 --- Coding with only total Sz conservation --- p.11 / Chapter 2.2.2 --- Coding by using translational symmetry --- p.12 / Chapter 2.2.3 --- Coding with H acting on spin configuration --- p.17 / Chapter 2.2.4 --- Coding on finding eigenvalues and eigenvectors --- p.20 / Chapter 2.3 --- Coding on calculating dynamic properties --- p.20 / Chapter 2.3.1 --- Coding on calculating thermal properties --- p.20 / Chapter 2.3.2 --- Coding on calculating other thermal property --- p.21 / Chapter 3 --- Finite temperature calculations on unfrustrated spin systems --- p.30 / Chapter 3.1 --- Introduction --- p.30 / Chapter 3.2 --- Finite temperature calculations --- p.33 / Chapter 3.2.1 --- Energy spectrum E(k) --- p.33 / Chapter 3.2.2 --- Internal energy (E) --- p.39 / Chapter 3.2.3 --- Heat capacity Cv --- p.42 / Chapter 3.2.4 --- Uniform susceptibility x --- p.45 / Chapter 3.2.5 --- Staggered magnetization mz+ --- p.47 / Chapter 3.3 --- Linear Spin Wave Theory --- p.48 / Chapter 3.3.1 --- Linear Spin Wave Theory at zero temperature --- p.48 / Chapter 3.3.2 --- Linear Spin Wave Theory at finite temperature --- p.54 / Chapter 3.4 --- Phase Transition --- p.57 / Chapter 4 --- Finite temperature calculations on frustrated systems --- p.62 / Chapter 4.1 --- Introduction --- p.62 / Chapter 4.2 --- Finite temperature calculations --- p.65 / Chapter 4.2.1 --- Energy spectrum E(k) --- p.65 / Chapter 4.2.2 --- Internal energy (E) --- p.68 / Chapter 4.2.3 --- Heat capacity Cv --- p.69 / Chapter 4.2.4 --- Uniform susceptibility x --- p.71 / Chapter 4.2.5 --- "Fourier transform of susceptibility S(qx,qy)" --- p.72 / Chapter 4.3 --- Linear Spin Wave Theory --- p.73 / Chapter 5 --- Finite Size Scaling --- p.78 / Chapter 5.1 --- Introduction --- p.78 / Chapter 5.2 --- Infinite unfrustrated system --- p.79 / Chapter 5.2.1 --- Ground state energy E0 --- p.79 / Chapter 5.2.2 --- Internal Energy (E) --- p.80 / Chapter 5.2.3 --- Staggered magnetization mz+ --- p.81 / Chapter 5.3 --- Infinite frustrated system --- p.83 / Chapter 5.3.1 --- Ground state energy E0 --- p.84 / Chapter 6 --- Comparisons between unfrustrated system and frustrated system --- p.87 / Chapter 6.1 --- Energy spectrum E(k) --- p.88 / Chapter 6.2 --- Internal energy (E) --- p.91 / Chapter 6.3 --- Heat capacity Cv --- p.92 / Chapter 6.4 --- Uniform susceptibility x --- p.93 / Chapter 7 --- Spin Lattice Relaxation l/T1 --- p.94 / Chapter 7.1 --- Introduction --- p.94 / Chapter 7.2 --- Spin temperature --- p.95 / Chapter 7.3 --- Experimental setup and its principle --- p.97 / Chapter 7.4 --- Numerical calculations --- p.102 / Chapter 8 --- Conclusion --- p.104 / Bibliography --- p.106 / Chapter A --- Method of moments --- p.109

In twistor theory the nonlinear graviton construction realises four-dimensional antiself- dual Einstein manifolds as Kodaira moduli spaces of rational curves in threedimensional complex manifolds. We establish a Newtonian analogue of this procedure, in which four-dimensional Newton-Cartan manifolds arise as Kodaira moduli spaces of rational curves with normal bundle O + O(2) in three-dimensional complex manifolds. The isomorphism class of the normal bundle is unstable with respect to general deformations of the complex structure, exhibiting a jump to the Gibbons- Hawking class of twistor spaces. We show how Newton-Cartan connections can be constructed on the moduli space by means of a splitting procedure augmented by an additional vector bundle on the twistor space which emerges when considering the Newtonian limit of Gibbons-Hawking manifolds. The Newtonian limit is thus established as a jumping phenomenon. Newtonian twistor theory is extended to dimensions three and five, where novel features emerge. In both cases we are able to construct Kodaira deformations of the flat models whose moduli spaces possess Galilean structures with torsion. In five dimensions we find that the canonical affine connection induced on the moduli space can possess anti-self-dual generalised Coriolis forces. We give examples of anti-self-dual Ricci-flat manifolds whose twistor spaces contain rational curves whose normal bundles suffer jumps to O(2 - k) + O(k) for arbitrarily large integers k, and we construct maps which portray these big-jumping twistor spaces as the resolutions of singular twistor spaces in canonical Gibbons-Hawking form. For k > 3 the moduli space itself is singular, arising as a variety in an ambient complex space. We explicitly construct Newtonian twistor spaces suffering similar jumps. Finally we prove several theorems relating the first-order and higher-order symmetry operators of the Schrödinger equation to tensors on Newton-Cartan backgrounds, defining a Schrödinger-Killing tensor for this purpose. We also explore the role of conformal symmetries in Newtonian twistor theory in three, four, and five dimensions.