Conformally Natural Extensions of Continuous Circle Maps
Year of Dissertation:
2012
Conformally natural and continuous extensions were originally introduced by Douady and Earle for circle homeomorphisms, and later by Abikoff, Earle and Mitra for continuous degree ±1 monotone circle maps. The first main result of this thesis shows that conformally natural and continuous extensions exist for all continuous circle maps. The second main result shows that if f is a continuous circle map and is M-quasisymmetric on some arc on the unit circle S1, then such an extension of f is locally K-quasiconformal on a neighborhood of the arc in the open unit disk D, where the neighborhood and K depend only on M.
Problems in Additive Number Theory
Year of Dissertation:
2009
The first chapter deals with the following problem: Let f (n) be a growth function, and A be a sequence with f (n) < an Uf (n), U constant. Under what conditions is it possible to construct another sequence
String Topology & Compactified Moduli Spaces
Year of Dissertation:
2010
The motivation behind this work is to solve the master equation dX = X*X in a chain complex which is a direct sum of homomorphism complexes of tensor powers of a chain complex P, where P computes H(LM,M), the S^1-equivariant homology of the free loop space LM of a manifold M, relative to constant loops. Here, we solve a modification of this equation: dX = X*X + A and suggest an avenue for modifying the solution of the second equation to obtain a solution of the master equation. The solution of the second equation is constructed by building a pseudomanifold of string diagrams which has prescribed boundary. The string topology construction describes the action of cellular chains of the pseudomanifold on P. Further, the pseudomanifold is homeomorphic to a compactification of the moduli space of Riemann surfaces. A second smaller compactification is defined over which string topology operations conjecturally extend.
Geometrical aspects of linear differential equations over compact Riemann surfaces with reductive differential Galois group
Author:
Camilo Sanabria Malagon
Year of Dissertation:
2010
Advisor:
Richard Churchill
Suppose L(y) = 0 is a linear differential equation with reductive Galois
Weakly Measurable Cardinals and Partial Near Supercompactness
Year of Dissertation:
2011
I will introduce a few new large cardinal concepts. A weakly measurable cardinal is a new large cardinal concept obtained by weakening the familiar concept of a measurable cardinal. Specifically, a cardinal κ is weakly measurable if for every collection A containing at most κ+ many subsets of κ, there exists a nonprincipal κ-complete filter on κ measuring all sets in A. Every measurable cardinal is weakly measurable, but a weakly measurable cardinal need not be measurable. Moreover, while the GCH cannot fail first at a measurable cardinal, I will show that it can fail first at a weakly measurable cardinal. More generally, if κ is measurable, then we can make its weak measurability indestructible by the forcing Add(κ, η) for all η while forcing the GCH to hold below κ. Nevertheless, I shall prove that weakly measurable cardinals and measurable cardinals are equiconsistent.
Reducibility, Degree Spectra, ans Lowness in Algebraic Structures
Year of Dissertation:
2012
This dissertation addresses questions in computable structure theory, which is a branch of mathematical logic hybridizing computability theory and the study of familiar mathematical structures. We focus on algebraic structures, which are standard topics of discussion among model theorists. The structures examined here are fields, graphs, trees under a predecessor function, and Boolean algebras.
Endomorphisms of n-dimensional projective space over function fields
Year of Dissertation:
2009
Let K=k(C) be the function field of a complete nonsingular curve C over an arbitrary field k. The main result states an endomorphism of 1-dimensional projective space over K is isotrivial if and only if it has potential good reduction at all places v of K. This generalizes results of Benedetto for polynomial maps on 1-dimensional projective space over K and Baker for arbitrary rational maps on 1-dimensional projective space over K. There are two proofs given. The first uses algebraic geometry and more specifically, geometric invariant theory. It is new even in the case of 1-dimensional projective space over K. The second proof, using non-archimedean analysis and dynamics, more directly generalizes proofs of Benedetto and Baker for the N=1 case. In addition, two applications for the result are given.
Holomorphic Motions and Extremal Annuli
Year of Dissertation:
2011
Advisor:
Frederick Gardiner
Holomorphic motions, soon after they were introduced, became an
The Minimal Resultant And Conductor for Self Maps of the Projective Line
Year of Dissertation:
2011
We develop and study the minimal resultant divisor of self-maps of the projective line over a number field or a function field and its relation to the conductor. The guiding focus is the exploration of a dynamical analog to Szpiro's Theorem (Theorem 5.1), which bounds the degree of the minimal discriminant of an elliptic curve over a function field in terms of the conductor. We also explore a question about Lattes maps: given an elliptic curve over a function field with semi-stable bad reduction, we show the associated Lattes maps for multiplication by n have unstable bad reduction for each n. We then study minimality and semi-stability, considering what conditions imply minimality and whether semi-stable models and presentations are minimal, proving results in the degree two case. We prove the singular reduction of a semi-stable presentation coincides with the bad reduction. Degree 2 maps in normal form with semi-stable bad reduction are used to construct a counterexample to a natural dynamical analog to Szpiro's Theorem. Finally, we consider the notion of "critical bad reduction," and show that a dynamical analog may still be possible using the locus of critical bad reduction to define the conductor.
Drawdowns, Drawups, and Their Applications
Year of Dissertation:
2010
Advisor:
Olympia Hadjiliadis
This thesis studies the probability characteristics of drawdown and drawup processes of general linear diffusions. The drawdown process is defined as the current drop from the running maximum, while the drawup process is defined as the current rise over the running minimum. Attention is drawn to the first hitting times of the drawdown and the drawup processes, also known as the drawdown and the drawup respectively, and their applications in managing financial risks and detecting abrupt changes in random processes. The probabilities that the drawdown of a units precedes the drawup of equal size are derived in a biased simple random walk model and a drifted Brownian motion model. It is then shown that there exists an analytical formula for the Laplace transform of the drawdown of a units when it precedes the drawup of b units. The above problem can be related to the arbitrage-free pricing of a digital option related to the drawdowns and the drawups. Several static and semi-static replications are developed to hedge the risk exposure of these options. Finally, we study the properties of the drawups as a means of detecting abrupt changes in random processes with multi-source observations. In particular, we study extensions of the cumulative sum (CUSUM) stopping rule, which is the drawup of the log-likelihood ratio process. It is shown that the N-CUSUM stopping rule is at least second-order asymptotically optimal as the meantime to the first false alarm tends to infinity.