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University of Colorado

1. Wakefield, Nathan Paul. Primitive Divisors in Generalized Iterations of Chebyshev Polynomials.

Degree: PhD, Mathematics, 2013, University of Colorado

URL: https://scholar.colorado.edu/math_gradetds/25

Let (<em>g_{i}</em>)<em>_{i}</em>_{ ≥1} be a sequence of Chebyshev polynomials, each with degree at least two, and define (<em>f_{i}</em>) <em>_{i}</em>_{ ≥1} by the following recursion: *f*_{1} = * g*_{1}, <em>f_{n}</em> = <em>g_{n}</em> ∘ <em> f_{n}</em>_{–1}, for *n* ≥ 2. Choose α ∈ [special characters omitted] such that {[special characters omitted](α) : *n* ≥ 1} is an infinite set. The main result is as follows: Let γ ∈ {0, ±1}, if <em>f_{ n}</em>(α) = [special characters omitted] is written in lowest terms, then for all but finitely many * n* > 0, the numerator, <em>A_{n}</em>, has a primitive divisor; that is, there is a prime *p* which divides <em> A_{n}</em> but does not divide <em>A_{i}</em> for any *i* < *n*.
In addition to the main result, several of the tools developed to prove the main result may be of interest.
A key component of the main result was the development of a generalization of canonical height. Namely: If [*f*] is a set of rational maps, all commuting with a common function *f*, and f = [special characters omitted] is a generalized iteration of rational maps formed by <em>f_{ n}</em>(*x*) = <em>g_{n}</em>(<em> f_{n}</em>_{–1}(*x*)) with <em> g_{i}</em> coming from [*f*], then there is a unique canonical height funtion *ĥ*_{f} : *K* → [special characters omitted] which is identical to the canonical height function associated to * f.*
Another key component of the main result was proving that under certain circumstances, being acted upon by a Chebyshev polynomial does not lead to significant differences between the size of the numerator and denominator of the result. Specifically, let γ ∈ {0, ±1, ±2} be fixed, and <em>g_{i}</em> be a sequence of Chebyshev polynomials. Let *f* given by the following recurrence *f*_{ 1}(*z*) = *g*_{1}(*z*), and <em>f_{i}</em> = <em>g_{i}</em>(<em> f_{i}</em>_{–1}(*z*)) for * i* ≥ 2. Pick any α ∈ [special characters omitted] with |α + γ| < 2, such that α + γ is not pre-periodic for one hence any Chebyshev polynomial. Write <em>f_{ n}</em>(α + γ) − γ = [special characters omitted] in lowest terms. Then limn→∞ logAn logBn =1. Finally, some areas of future research are discussed.
*Advisors/Committee Members: Su-Ion Ih, Katherine Stange, Robert Tubbs, Eric Stade, Juan Restrepo.*

Subjects/Keywords: Arithmetic Dynamics; Chebyshev; Generalized Iteration; Primitive Divisors; Mathematics

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^{th} Edition):

Wakefield, N. P. (2013). Primitive Divisors in Generalized Iterations of Chebyshev Polynomials. (Doctoral Dissertation). University of Colorado. Retrieved from https://scholar.colorado.edu/math_gradetds/25

Chicago Manual of Style (16^{th} Edition):

Wakefield, Nathan Paul. “Primitive Divisors in Generalized Iterations of Chebyshev Polynomials.” 2013. Doctoral Dissertation, University of Colorado. Accessed November 29, 2020. https://scholar.colorado.edu/math_gradetds/25.

MLA Handbook (7^{th} Edition):

Wakefield, Nathan Paul. “Primitive Divisors in Generalized Iterations of Chebyshev Polynomials.” 2013. Web. 29 Nov 2020.

Vancouver:

Wakefield NP. Primitive Divisors in Generalized Iterations of Chebyshev Polynomials. [Internet] [Doctoral dissertation]. University of Colorado; 2013. [cited 2020 Nov 29]. Available from: https://scholar.colorado.edu/math_gradetds/25.

Council of Science Editors:

Wakefield NP. Primitive Divisors in Generalized Iterations of Chebyshev Polynomials. [Doctoral Dissertation]. University of Colorado; 2013. Available from: https://scholar.colorado.edu/math_gradetds/25

University of Colorado

2. Havasi, Krisztián. Geometric Realization of Strata in the Boundary of the Intermediate Jacobian Locus.

Degree: PhD, Mathematics, 2016, University of Colorado

URL: https://scholar.colorado.edu/math_gradetds/42

In this thesis we describe intermediate Jacobians of threefolds obtained from singular cubic threefolds. By this we mean two things. First, we describe the intermediate Jacobian of a desingularization of a cubic threefold with isolated singularities. Second, we describe limits of intermediate Jacobians of smooth cubic threefolds, as the family of cubic threefolds acquires isolated singularities. In regards to the first question, generalizing a result of Clemens – Griffiths we show specifically that the intermediate Jacobian of a distinguished desingularization of a cubic threefold with a single singularity of type A_{3} is the Jacobian of the normalization of an associated complete intersection curve in ℙ^{3}, the so called (2,3)-curve. In regards to degenerations, we describe how the limit intermediate Jacobian, under certain conditions, can be described as a semi-abelian variety as the extension of a torus by the finite quotient of the product of Jacobians of curves, where one of the curves is the normalization of the (2,3)-curve associated to the cubic threefold and a choice of singularity, and the other curves are so-called tails arising from stable reduction of plane curve singularities.
*Advisors/Committee Members: Sebastian Casalaina-Martin, Jonathan Wise, Su-ion Ih, Jeff Achter, Samouil Molcho.*

Subjects/Keywords: complex geometry; cubic threefold; degenerate intermediate Jacobian; degenerate Prym variety; Geometry and Topology

Record Details Similar Records

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^{th} Edition):

Havasi, K. (2016). Geometric Realization of Strata in the Boundary of the Intermediate Jacobian Locus. (Doctoral Dissertation). University of Colorado. Retrieved from https://scholar.colorado.edu/math_gradetds/42

Chicago Manual of Style (16^{th} Edition):

Havasi, Krisztián. “Geometric Realization of Strata in the Boundary of the Intermediate Jacobian Locus.” 2016. Doctoral Dissertation, University of Colorado. Accessed November 29, 2020. https://scholar.colorado.edu/math_gradetds/42.

MLA Handbook (7^{th} Edition):

Havasi, Krisztián. “Geometric Realization of Strata in the Boundary of the Intermediate Jacobian Locus.” 2016. Web. 29 Nov 2020.

Vancouver:

Havasi K. Geometric Realization of Strata in the Boundary of the Intermediate Jacobian Locus. [Internet] [Doctoral dissertation]. University of Colorado; 2016. [cited 2020 Nov 29]. Available from: https://scholar.colorado.edu/math_gradetds/42.

Council of Science Editors:

Havasi K. Geometric Realization of Strata in the Boundary of the Intermediate Jacobian Locus. [Doctoral Dissertation]. University of Colorado; 2016. Available from: https://scholar.colorado.edu/math_gradetds/42