# Alumni Dissertations

• ### 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.

• ### 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.