### Browsing by Subject "Algebraic topology"

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Item Applications of Semiclassical Theory in Statistical and Quantum Mechanics(2018-06) Janas, MichaelShow more Since the initiation of quantum theory in the early 20th century, semiclassical methods have been a perenniel source of insight into a diverse range of phenomena. In spite of this history, however, there remain interesting and insightful applications of semiclassical theory to physics. This thesis advances this programme in several directions. First, we consider the statistical mechanics of multivalent 1D Coulomb gases and demonstrate how the semiclassical WKB method may be used to expose its thermodynamic properties. In doing so, we develop ideas from algebraic topology and complex Riemann surfaces. Moving to quantum theory proper, these tools are applied fruitfully to the phenomenon of spin tunneling oscillations in magnetic molecules with large instrinsic spin. Moving away from the WKB approximation, these ideas from complex analysis also proved crucial in exposing universal finite-size scaling effects in 1D lattice systems such as the Su-Schrieffer-Heeger model of polyacetylene and the Kitaev chain. Finally, we end by considering the the weak noise theory of the KPZ equation and thereby discover a novel phase transition in its large deviation statistics.Show more Item The homology of the kernel space of the Thom spectrum in low degrees(2024) Huttinga, ZaneShow more I present an investigation of the multiplicative structure of H_*(SL_1 MU; F_2) as an algebra over F_2. This includes a list of generators in low degrees and the multiplication table for these generators. This ring has applications in topology related to orientability of vector bundles, as it is closely tied to the homology of the spectrum sl_1 MU. The ring H_*(SL_1 MU) is closely tied to the Hopf ring H_*(MU_2*) introduced by Ravenel and Wilson in 1977. All of the available multiplicative information on H_*(SL_1 MU) descends from H_*(MU_2*), and accordingly, I present a complete algebraic description of this Hopf ring.Show more Item A Semiclassical Theory on Complex Manifolds with Applications in Statistical Physics and Quantum Mechanics(2016-08) Gulden, TobiasShow more Increased interest in non-Hermitian quantum systems calls for the development of efficient methods to treat these. This interest was sparked by the introduction of PT-symmetry and the study of mathematical mappings which map conventional statistical or quantum mechanics onto non-Hermitian quantum operators. One of the most common methods in quantum mechanics is the semiclassial approximation which requires integration along trajectories that solve classical equations of motion. However in non-Hermitian systems these solutions are rarely attainable. We borrow concepts from algebraic topology to develop methods to avoid solving the equations of motion and avoid straightforward integration altogether. We apply these methods to solve the semiclassical problem for three largely dierent systems and demonstrate their usefulness for Hermitian and non-Hermitian systems alike.Show more