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- 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
- ItemSome advances in a conjecture of Watkins and an analogue over function fields(2023) Caro Reyes, Jerson; Pastén Vásquez, Héctor; Pontificia Universidad Católica de Chile. Facultad de MatemáticasOur results are divided into two main parts, both related to a conjecture by Watkins. In 2002, Watkins conjectured that the rank of an elliptic curve defined over Q is at most the 2-adic valuation of its modular degree. The first part is related to presenting some approaches to Watkins’s conjecture in its original version. We prove this conjecture for semistable elliptic curves having exactly one rational point of order 2, provided that they have an odd number of primes of non-split multiplicative reduction or no primes of split multiplicative reduction. In addition, we show that this conjecture is satisfied when E is any quadratic twist of an elliptic curve with non-trivial rational 2-torsion and prime power conductor, in particular, for the congruent number elliptic curves. In the second part, we consider the analogous problem over function fields of positive characteristic, and we prove it in several cases. More precisely, every modular semistable elliptic curve over Fq(T) after extending constant scalars and every quadratic twist of a modular elliptic curve over Fq(T) by a polynomial with sufficiently many prime factors satisfy this version of Watkins’s conjecture. Additionally, we prove the analogue of Watkins’s conjecture for a well-known family of elliptic curves with unbounded rank due to Ulmer. In addition, we include a final appendix describing joint work with Hector Pasten [16] on a generalization of the Chabauty-Coleman bound for surfaces. While this is not directly related to the core of the thesis, it is a report on work that was performed during my time as a Ph.D. student.