# Alumni Dissertations

• ### 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 $S$.

• ### Motivic integration over nilpotent structures

Author:
Andrew Stout
Year of Dissertation:
2014
Program:
Mathematics
Hans Schoutens
Abstract:

This thesis concerns developing the notion of Motivic Integration in such a way that it captures infinitesimal information yet reduces to the classical notion of motivic integration for reduced schemes. Moreover, I extend the notion of Motivic Integration from a discrete valuation ring to any complete Noetherian ring with residue field $\kappa$, where $\kappa$ is any field. Schoutens' functorial approach (as opposed to the traditional model theoretic approach) allows for some very general notions of motivic integration. However, the central focus is on using this general framework to study generically smooth schemes, then non-reduced schemes, and then, finally, formal schemes. Finally, a computational approach via Sage for computing the equations defining affine arc spaces is introduced and implemented.

• ### Stable Commutator Length in Amalgamated Free Products

Author:
Timothy Susse
Year of Dissertation:
2014
Program:
Mathematics
Jason Behrstock
Abstract:

We show that stable commutator length is rational on free products of free Abelian groups amalgamated over Zk, a class of groups containing the fundamental groups of all torus knot complements. We consider a geometric model for these groups and parameterize all surfaces with specified boundary mapping to this space. Using this work we provide a topological algorithm to compute stable commutator length in these groups. We then use the combinatorics of this algorithm to prove that for a word w in the (p, q)-torus knot complement, scl(w) is quasirational in p and q. Finally, we analyze central extensions, and prove that under certain conditions the projection map preserves stable commutator length.

• ### 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
Amotz Bar-Noy
Abstract:

Mobile sensors are located on a barrier represented by a line segment, and each sensor has a single energy source that can be used for both moving and sensing. Sensors may move once to their desired destinations and then coverage/communication is commenced. The sensors are collectively required to cover the barrier or in the communication scenario set up a chain of communication from endpoint to endpoint. A sensor consumes energy in movement in proportion to distance traveled, and it expends energy per time unit for sensing in direct proportion to its radius raised to a constant exponent. The first focus is of energy efficient coverage. A solution is sought which minimizes the sum of energy expended by all sensors while guaranteeing coverage for a predetermined amount of time. The objective of minimizing the maximum energy expended by any one sensor is also considered. The dual model is then studied. Sensors are equipped with batteries and a solution is sought which maximizes the coverage lifetime of the network, i.e. the minimum lifetime of any sensor. In both of these models, the variant where sensors are equipt with predetermined radii is also examined. Lastly, the problem of maximizing the lifetime of a wireless connection between a transmitter and a receiver using mobile relays is considered. These problems are mainly examined from the point of view of approximation algorithms due to the hardness of many of them.

• ### 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 important subject in complex analysis. It is now an important tool in the study of complex dynamical systems and in the study of Teichmuller theory. This thesis serves on two purposes: an expository of the past developments and a discovery of new theories. First, I give an expository account of Slodkowski's theorem based on the proof given by Chirka. Then I present a result about infinitesimal holomorphic motions. I prove the $|\epsilon \log\epsilon|$ modulus of continuity for any infinitesimal holomorphic motion. This proof is a very well application of Schwarz's lemma and the estimate of Agard's formula for the hyperbolic metric on the thrice punctured sphere. One application of this result is that, after the integration of an infinitesimal holomorphic motion, it leads to the Holder continuity property of a quasiconformal homeomorphism. This will be presented in Chapter 3. Second, I compare the proofs given by both Slodkowski and Chirka. Then I construct a different extension of a holomorphic motion in the frame work of Slodkowsk's proof by using the method in Chirka's proof. This gives some opportunity for me to discuss the uniqueness in the extension problem for a holomorphic motion. This will be presented in Chapter 4. Third, I discuss the universal holomorphic motion for a closed subset of the Riemann sphere and the lifting property in the Teichmuller theory. One application of this discussion is the proof of the coincidence of Teichmuller's metric and Kobayashi's metric, a result due to Royden and Gardiner, given by Earle, Kra, and Krushkal by using Slodkowski's theorem. This will be presented in Chapters 5 and 6. Fourth, I study the complex structure of the universal asymptotically conformal Teichmuller space. I give a direct and new proof of the coincidence of Teichmuller's metric and Kobayashi's metric on the universal asymptotically conformal Teichmuller space, a result previously proved by Earle, Gardiner, and Lakic. The main technique that I have used in this proof is Strebel's frame mapping theorem. This will be presented in Chapter 7. Finally, in Chapter 8, I study extremal annuli on a Riemann sphere with four points removed. By using the measurable foliation theory, the Weierstrass P-function, and the variation formula for the modulus of an annulus, I prove that the Mori annulus maximize the modulus for the two army problem in the chordal distance on the Riemann sphere. Gardiner and Masur's minimum axis is also discussed in this chapter. Most of the results in this thesis have been published in several research papers jointly with Fred Gardiner, Jun Hu, Yunping Jiang, and Sudeb Mitra.