# Alumni Dissertations

• ### Inverse limits of models of set theory and the large cardinal hierarchy near a high-jump cardinal

Author:
Norman Perlmutter
Year of Dissertation:
2013
Program:
Mathematics
Joel Hamkins
Abstract:

This dissertation consists of two chapters, each of which investigates a topic in set theory, more specifically in the research area of forcing and large cardinals. The two chapters are independent of each other. The first chapter analyzes the existence, structure, and preservation by forcing of inverse limits of inverse-directed systems in the category of elementary embeddings and models of set theory. Although direct limits of directed systems in this category are pervasive in the set-theoretic literature, the inverse limits in this same category have seen less study. I have made progress towards characterizing the existence and structure of these inverse limits. Some of the most important results are as follows. An inverse limit exists if and only if a natural source exists. If the inverse limit exists, then it is given by either the entire thread class or by a rank-initial segment of the thread class. Given sufficient large cardinal hypotheses, it is consistent that there are systems with no inverse limit, systems with inverse limit given by the entire thread class, and systems with inverse limit given by a proper subset of the thread class. Inverse limits are preserved by forcing in both directions under fairly general assumptions but not in all cases. Prikry forcing and iterated Prikry forcing are important techniques for constructing some of the examples in this chapter. The second chapter analyzes the hierarchy of the large cardinals between a supercompact cardinal and an almost-huge cardinal, including in particular high-jump cardinals. I organize the large cardinals in this region by consistency strength and implicational strength. I also prove some results relating high-jump cardinals to forcing. A high-jump cardinal is the critical point of an elementary embedding j: V --> M such that M is closed under sequences of length equal to the clearance of the embedding. This clearance is defined as the supremum, over all functions f from κ to κ, of j(f)(κ). Two of the most important results in the chapter are as follows. A Vopenka cardinal is equivalent to a Woodin-for-supercompactness cardinal. There are no excessively hypercompact cardinals.

• ### String Topology & Compactified Moduli Spaces

Author:
Katherine Poirier
Year of Dissertation:
2010
Program:
Mathematics
Dennis Sullivan
Abstract:

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
Program:
Mathematics
Richard Churchill
Abstract:

Suppose L(y) = 0 is a linear differential equation with reductive Galois group over the function field of a compact Riemann surface. We prove that any solution to the equation can be written as a product of a solution to a first order equation and a solution to the pullback of an equation of a special form (a "standard equation"). We classify standard equations using ruled surfaces. We relate the symmetries of L(y) = 0 to the outer-automorphisms of the differential Galois group.

• ### 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. A cardinal κ is nearly θ-supercompact if for every A that's a subset of θ, there exists a transitive M |= ZFC- closed under <κ sequences having the subset A and the cardinals κ and θ as elements, a transitive N, and an elementary embedding j: M -> N with critical point κ such that j(κ) > θ and j''θ is in N. This concept strictly refines the θ-supercompactness hierarchy as every θ-supercompact cardinal is nearly θ-supercompact, and every nearly 2θ-supercompact cardinal κ is θ-supercompact. Moreover, if κ is a θ-supercompact cardinal for some θ such that θ = θ, we can move to a forcing extension preserving all cardinals below θ++ where κ remains θ-supercompact but is not nearly θ+-supercompact. I will also show that if κ is nearly θ-supercompact for some θ ≥ 2κ such that θ = θ, then there exists a forcing extension preserving all cardinals at or above κ where κ is nearly θ-supercompact but not measurable. These types of large cardinals also come equipped with a nontrivial indestructibility result, and I will prove that if κ is nearly θ-supercompact for some θ ≥ κ such that θ = θ, then there is a forcing extension where its near θ-supercompactness is preserved and indestructible by any further <κ-directed closed θ-c.c. forcing of size at most θ. Finally, these cardinals have high consistency strength. Specifically, I will show that if κ is nearly θ-supercompact for some θ ≥ κ+ for which θ = θ, then AD holds in L(R). In particular, if κ is nearly κ+-supercompact and 2κ = κ+, then AD holds in L(R).

• ### Special Representations, Nathanson's Lambda Sequences and Explicit Bounds

Author:
Satyanand Singh
Year of Dissertation:
2014
Program:
Mathematics
Melvyn Nathanson
Abstract:

{Let $X$ be a group with identity $e$, we define $A$ as an infinite set of generators for $X$, and let $(X,d)$ be the metric space with word length $d_{A}$ induced by $A$. Nathanson showed that if $P$ is a nonempty finite set of prime numbers and $A$ is the set of positive integers whose prime factors all belong to $P$, then the metric space $({\bf{Z}},d_{A})$ has infinite diameter. Nathanson also studied the $\lambda_{A}(h)$ sequences, where $\lambda_{A}(h)$ is defined as the smallest positive integer $y$ with $d_{A}(e,y)=h$, and he posed the problem to compute $\lambda_{A}(h)$ and estimate its growth rate. We will give explicit forms for $\lambda_{p}(h)$ for any fixed odd integer $p>1$. We will also solve the open problems of computing the term $\lambda_{2,3}(4)$, provide an explicit lower bound for $\lambda_{2,3}(h)$ and classifying $\lambda_{2,p}(h)$ for $p>1$ any odd integer and $h\in\{1,2,3\}$. }

• ### 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. For a computable field F, the splitting set SF of F is the set of polynomials in F[X] which factor over F, and the root set RF of F is the set of polynomials in F[X] which have a root in F. Results of Frohlich and Shepherdson from 1956 imply that for a computable field F, the splitting set SF and the root set RF are Turing-equivalent. Much more recently, in 2010, R. Miller showed that for algebraic fields, if we use a finer measure, the root set actually has slightly higher complexity: for algebraic fields F, it is always the case that SF1 RF, but there are algebraic fields F where we don't have RF1 SF. In the first chapter, we compare the splitting set and the root set of a computable algebraic field under a different reduction: the bounded Turing (bT) reduction. We construct a computable algebraic field for which we don;t have RF1 SF. We also define a Rabin embedding g of a field into its algebraic closure, and for a computable algebraic field F, we compare the relative complexities of RF, SF, and g(F) under m–reducibility and under bT–reducibility. Work by R. Miller in 2009 proved several theorems about algebraic fields and computable categoricity. Also in 2009, A. Frolov, I. Kalimullin, and R. Miller proved some results about the degree spectrum of an algebraic field when viewed as a subfield of its algebraic closure. In the second chapter, we show that the same computable categoricity results also hold for finite-branching trees under the predecessor function and for connected, finite-valence, pointed graphs, and we show that the degree spectrum results do not hold for these trees and graphs. We also offer an explanation for why the degree spectrum results distinguish these classes of structures: although all three structures are algebraic structures, the fields are what we call effectively algebraic. Every lown Boolean algebra, for 1 ≤ n ≤ 4, is isomorphic to a computable Boolean algebra. It is not yet known whether the same is true for n > 4. However, it is known that there exists a low5 subalgebra of the computable atomless Boolean algebra which, when viewed as a relation on the computable atomless Boolean algebra, does not have a computable copy. In the third chapter, we adapt the proof of this recent result to show that there exists a low4 subalgebra of the computable atomless Boolean algebra B which, when viewed as a relation on B, has no computable copy. This result provides a sharp contrast with the one which shows that every low4 Boolean algebra has a computable copy. That is, the spectrum of the subalgebra as a unary relation can contain a low4 degree without containing the degree 0, even though no spectrum of a Boolean algebra (viewed as a structure) can do the same. We also point out that unlike Boolean algebras as structures, which cannot have nth–jump degree above 0(n), subalgebras of B considered as relations on B can have nth–jump degree strictly bigger than 0(n).

• ### 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.

• ### Lean, Green, and Lifetime Maximizing Sensor Deployment on a Barrier

Author:
Peter Terlecky
Year of Dissertation:
2014
Program:
Mathematics