Symmetry analysis of geometries in general relativity and mathematical models in quantitative finance

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This thesis may be divided into two themes. The first, consists of a mathematical analysis of selected models in cosmology, while the second, is in the field of quantitative finance. The pp-wave spacetime, Bianchi I spacetime, and the Bianchi II spacetime are the three universes that we examine. The latter two are studied in the framework of f(R) theory of gravity, a plausible substitute for general relativity and a solution to the dark energy problem. We apply the Killing and homothetic vector fields to divide the spacetimes into classes or categories. Subsequently, potential functions are established using the geometry of the point symmetries of the space, while Noether’s theorem provides the first integrals connected to each isometry. We take advantage of the geometric fact that the homothetic algebra of spacetimes yields the Noether point symmetries of geodesic Lagrangians. For the Bianchi spacetimes, the Wheeler-DeWitt equations are derived by the quantisation of the spacetime Lagrangians, and from the Lie point symmetries it admits, we sequentially find invariant solutions to solve for the universe’s wave function. Finally, exact solutions to the field equations are also found. The research comprising of the second theme, investigates two nonlinear partial differential equations used in derivative pricing for financial markets. The point symmetries, invariant solutions, and conversation laws of these equations are found. In our analysis, we vary certain variables that change the nonlinearity of the models and thus give us unique symmetries and solutions. With various parameter settings, graphical solutions are investigated.
A thesis submitted in fulfilment of the requirements for the degree of Doctor of Philosophy to the Faculty of Science, School of Mathematics, University of the Witwatersrand, Johannesburg, 2023
Symmetry analysis, Geometries