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- ItemThe noncommutative geometry of the Landau Hamiltonian(2024) Sandoval, Maximiliano; De Nittis, Giuseppe; Pontificia Universidad Católica de Chile. Facultad de MatemáticasThis 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.
- ItemFibred non-hyperbolic quadratic families(2024) Domínguez Calderón, Igsyl; Ponce Acevedo, Mario; Pontificia Universidad Católica de Chile. Facultad de MatemáticasThe aim of this thesis is two-folding. In the initial instance, we have made signifIcant progress in the problem of density of hyperbolic components within the context of fibred quadratic polynomial dynamics by demonstrating the existence of robust non- hyperbolic fibred quadratic polynomials. Secondly, we present a more complex class of invariant sets that are distinct from the invariant curves for fibred polynomial dynamics, called multi-curves. Furthermore, a construction for multi-curves in quadratic polynomial dynamics is shown, resulting in the attainment of not only invariant multi-curves, but also with the characteristic of being attracting.
- ItemC*-algebric methods for transport phenomena(2023) Polo Ojito, Danilo; De Nittis, Giuseppe; Pontificia Universidad Católica de Chile. Facultad de Matemáticas
- ItemOn the geography of 3-folds via asymptotic behavior of invariants(2023) Torres Nova, Yerko Alejandro; Urzúa Elia, Giancarlo A.; Pontificia Universidad Católica de Chile. Facultad de MatemáticasRoughly speaking, the problem of geography asks for the existence of varieties of general type after we fix some invariants. In dimension 1, where we fix the genus, the geography question is trivial, but already in dimension 2 it becomes a hard problem in general. In higher dimensions, this problem is essentially wide open. In this paper, we focus on geography in dimension 3. We generalize the techniques which compare the geography of surfaces with the geography of arrangements of curves via asymptotic constructions. In dimension 2 this involves resolutions of cyclic quotient singularities and a certain asymptotic behavior of the associated Dedekind sums and continued fractions. We discuss the general situation with emphasis in dimension 3, analyzing the singularities and various resolutions that show up, and proving results about the asymptotic behavior of the invariants we fix.
- ItemEssential minimum in families(2023) Morales Inostroza, Marcos; Kiwi Krauskopf, Jan Beno; Sombra, Martín; Pontificia Universidad Católica de Chile. Facultad de Matemáticas