# Alumni Dissertations and Theses

• ### Conformally Natural Extensions of Continuous Circle Maps

Author:
Oleg Muzician
Year of Dissertation:
2012
Program:
Mathematics
Jun Hu
Abstract:

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

Author:
Brooke Orosz
Year of Dissertation:
2009
Program:
Mathematics
Melvyn Nathanson
Abstract:

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 with bk asymptotically equal to Bf (k), which has A as a subsequence? The next two chapters deal with the possible sizes of generalized sum sets on finite sets of integers. The final chapter discusses counting relatively prime subsets of the natural numbers.

• ### 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.

• ### 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.

• ### 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).

• ### 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 $h\in\{1,2,3\}$. }