# Alumni Dissertations

• ### Geometrical aspects of linear differential equations over compact Riemann surfaces with reductive differential Galois group

Author:
Camilo Sanabria Malagon
Year of Dissertation:
2010
Program:
Mathematics
Richard Churchill
Abstract:

Suppose L(y) = 0 is a linear differential equation with reductive Galois

• ### Weakly Measurable Cardinals and Partial Near Supercompactness

Author:
Jason Schanker
Year of Dissertation:
2011
Program:
Mathematics
Joel Hamkins
Abstract:

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

Author:
Rebecca Steiner
Year of Dissertation:
2012
Program:
Mathematics
Russell Miller
Abstract:

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.

• ### Dynamical Shafarevich results for rational maps.

Author:
Brian Stout
Year of Dissertation:
2013
Program:
Mathematics
Calyton Petsche
Abstract:

Given a number field $K$ and a finite set $S$ of places of $K$, this dissertation studies rational maps with prescribed good reduction at every place $v\not\in S$. The first result shows that the set of all quadratic rational maps with the standard notion of good reduction outside $S$ is Zariski dense in the moduli space $\Mcal_2$. The second result shows that if the notion of good reduction is strengthened by requiring a double unramified fixed point structure or an unramified two cycle, then one obtains a non-Zariksi density statement. The next result proves the existence of global minimal models of endomorphisms on $\PP^n$ defined over the fractional field of principal ideal domain. This result is used to prove the last main theorem- the finiteness of twists of a rational maps on $\PP^n$ over $K$ with good reduction outside $S$.

• ### Non-simple Closed Geodesics on 2-Orbifolds

Author:
Robert Suzzi Valli
Year of Dissertation:
2013
Program:
Mathematics
Ara Basmajian
Abstract:

Given a Fuchsian group Γ, that is, a discrete subgroup of the group of orientation-preserving isometries of the hyperbolic plane H, the quotient H/Γ is a 2-orbifold. If $Gamma$ contains torsion then the resulting 2-orbifold contains cone points corresponding to the elliptic fixed points. In this thesis we focus on minimal length non-simple closed geodesics on 2-orbifolds. Nakanishi, Pommerenke and Purzitsky discovered the shortest non-simple closed geodesic on a 2-orbifold, which passes through a cone point of the orbifold. This raises questions about minimal length non-simple closed geodesics disjoint from the cone points. We explore once self-intersecting closed geodesics disjoint from the cone points of the orbifold, called figure eight geodesics. Using fundamental domains and basic hyperbolic trigonometry we identify and classify all figure eight geodesics on triangle group orbifolds. This classification allows us to find the shortest figure eight geodesic on a triangle group orbifold. We then generalize to find the shortest figure eight geodesic on a 2-orbifold without cone points of order two.

• ### Endomorphisms of n-dimensional projective space over function fields

Author:
Michael Tepper
Year of Dissertation:
2009
Program:
Mathematics
Lucien Szpiro
Abstract:

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.

• ### Trees, Prisms and a Quillen Model Structure on Prismatic Sets

Author:
Louis Thrall
Year of Dissertation:
2013
Program:
Mathematics
Martin Bendersky
Abstract:

We define a category which we call the category of prisms, P. This category interpolates between the simplex category, &# 916; and the box category ☐ . The way in which we interpolate these two categories is considering categories with a tensor functor and a cone functor. It turns out that the objects of the category P are in one to one correspondence with planer rooted trees. Furthermore the morphisms of this category may be defined as certain combinatorial decorations on certain trees. We then show that that the category P is a test category automatically giving a Quillen model structure on the presheaves on P.

• ### Holomorphic Motions and Extremal Annuli

Author:
Zhe Wang
Year of Dissertation:
2011
Program:
Mathematics
Frederick Gardiner
Abstract:

Holomorphic motions, soon after they were introduced, became an

• ### The Minimal Resultant And Conductor for Self Maps of the Projective Line

Author:
Phillip Williams
Year of Dissertation:
2011
Program:
Mathematics
Lucien Szpiro
Abstract:

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

Author:
Hongzhong Zhang
Year of Dissertation:
2010
Program:
Mathematics
Abstract:

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.