The noncommutative geometry of the Landau Hamiltonian
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Date
2024
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Abstract
This dissertation will present three works in the areas of noncommutative geometry, the study of the Landau Hamiltonian, and the study of rational noncommutative tori making use of complex geometry and theta functions. The first two works focus primary on extending Bellissard’s work on the noncommutative geometry of the Hall Effect to the case where the medium is continuous case making use of a novel Dirac Operator, closely related to the isotropic quantum harmonic oscillator. We study the resulting spectral triple and its properties as a noncommutative space. Notably we provide proofs for the first and second Connes formulas for this spectral triple. The third work deals with the study of a pair of dual representations of rational noncommutative tori with rational parameters θ and θ −1, how we can naturally construct a vector bundles on a complex tori from to them, and hint how this duality is a manifestation of the Fourier-Mukai-Nahm transform.
Description
Tesis (Doctor in Mathematics)--Pontificia Universidad Católica de Chile, 2024.
Keywords
Hamiltoniano de Landau, Triples espectrales, Geometría no conmutativa