# Alumni Dissertations and Theses

• ### 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.

• ### ON POLYNOMIAL ROOTS APPROXIMATION VIA DOMINANT EIGENSPACES AND ISOLATION OF REAL ROOTS

Author:
OMAR RETAMOSO URBANO
Year of Dissertation:
2015
Program:
Mathematics
VICTOR PAN
Abstract:

Finding the roots of a given polynomial is a very old and noble problem in mathematics and computational mathematics. For about 4,000 years, various approaches had been proposed to solve this problem (see cite{FC99}). In 1824, Niels Abel showed that there existed polynomials of degree five, whose roots could not be expressed using radicals and arithmetic operations through their coefficients. Here is an example of such polynomials:$$x^5-4x-2.$$ Thus we must resort to iterative methods to approximate the roots of a polynomial given with its coefficients. There are many algorithms that approximate the roots of a polynomial(see cite{B40}, cite{B68}, cite{MN93}, cite{MN97}, cite{MN99}, cite{MN02}, cite{MN07}). As important examples we cite Quadtree (Weyl's) Construction and Newton's Iteration (see cite{P00a}). Some of the algorithms have as their goal to output a single root, for example, the absolutely largest root. Some other algorithms aim to output a subset of all the roots of the given polynomial, for example, all the roots within a fixed region on the complex plane. In many applications (e.g., algebraic geometric optimization), only the real roots are of interest, and they can be much less numerous than all the roots of the polynomial (see cite{MP13}). Nevertheless, the best numerical subroutines, such as MPSolve 2.0 cite{BF00}, Eigensolve cite{F02}, and MPsolve 3.0 cite{BR14}, approximate all real roots about as fast and as slow as all complex roots. The purpose of this thesis is to find real roots of a given polynomial effectively and quickly, this is accomplished by separating real roots from the other roots of the given polynomial and by finding roots which are clustered and absolutely dominant. We use matrix functions throughout this thesis to achieve this goal. One of the approaches is to approximate the roots of a polynomial $p(x)$ by approximating the eigenvalues of its associated companion matrix $C_{p}$. This takes advantage of using the well-known numerical matrix methods for the eigenvalues. This dissertation is organized as follows. Chapter 1 is devoted to brief history and modern applications of Polynomial root-finding, definitions, preliminary results, basic theorems, and randomized matrix computations. In Chapter 2, we present our Basic Algorithms and combine them with repeated squaring to approximate the absolutely largest roots as well as the roots closest to a selected complex point. We recall the matrix sign function and apply it to eigen-solving. We cover its computation and adjust it to real eigen-solving. In Chapter 3, we present a "matrix free" algorithm to isolate and approximate real roots of a given polynomial. We use a Cayley map followed by Dandelin's (Lobachevsky's, Gr"{a}ffe's) iteration. This is in part based on the fact that we have at hand good and efficient algorithms to approximate roots of a polynomial having only real roots (for instance the modified Laguerre's algorithm of [DJLZ97]). The idea is to extract (approximately) from the image of the given polynomial (via compositions of rational functions) a factor whose roots are all real, which can be solved using modified Laguerre's algorithm, so we can output good approximations of the real roots of the given polynomial. In Chapter 4, we present an algorithm based on a matrix version of the Cayley map used in Chapter 3. As our input, we consider the companion matrix of a given polynomial. The Cayley map and selected rational functions are treated as matrix functions. Via composition of matrix functions we generate and approximate the eigenspace associated with the real eigenvalues of the companion matrix, and then we readily approximate the real eigenvalues of the companion matrix of the given polynomial. To simplify the algorithm and to avoid numerical issues appearing in computation of the high powers of matrices, we use factorization of $P^{k}-P^{-k}$ as the product $prod_{i=0}^{k-1}(P-omega_k^iP^{-1})$ where $omega_k=exp(2pisqrt {-1}/k)$ is a primitive $k$th root of unity.

• ### On String Topology Operations and Algebraic Structures on Hochschild Complexes

Author:
Manuel Rivera
Year of Dissertation:
2015
Program:
Mathematics
Dennis Sullivan
Abstract:

The field of string topology is concerned with the algebraic structure of spaces of paths and loops on a manifold. It was born with Chas and Sullivan's observation of the fact that the intersection product on the homology of a smooth manifold $M$ can be combined with the concatenation product on the homology of the based loop space on $M$ to obtain a new product on the homology of $LM$, the space of free loops on $M$. Since then, a vast family of operations on the homology of $LM$ have been discovered. In this thesis we focus our attention on a non trivial coproduct of degree $1-\text{dim}(M)$ on the homology of $LM$ modulo constant loops. This coproduct was described by Sullivan on chains on general position and by Goresky and Hingston in a Morse theory context. We give a Thom-Pontryagin type description for the coproduct. Using this description we show that the resulting coalgebra is an invariant on the oriented homotopy type of the underlying manifold. The coproduct together with the loop product induce an involutive Lie bialgebra structure on the $S^1$-equivariant homology of $LM$ modulo constant loops. It follows from our argument that this structure is an oriented homotopy invariant as well. There is also an algebraic theory of string topology which is concerned with the structure of Hochschild complexes of DG Frobenius algebras and their homotopy versions. We make several observations about the algebraic theory around products, coproducts and their compatibilities. In particular, we describe a $BV$-coalgebra structure on the coHochschild complex of a DG cocommutative Frobenius coalgebra. Some conjectures and partial results regarding homotopy versions of this structure are discussed. Finally, we explain how Poincaré duality may be incorporated into Chen's theory of iterated integrals to relate the geometrically constructed string topology operations to algebraic structures on Hochschild complexes.

• ### 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 $S$.