# Alumni Dissertations and Theses

• ### On the Dynamics of Quasi-Self-Matings of Generalized Starlike Complex Quadratics and the Structure of the Mated Julia Sets

Author:
Ross Flek
Year of Dissertation:
2009
Program:
Mathematics
Linda Keen
Abstract:

It has been shown that, in many cases, Julia sets of complex polynomials can be "glued" together to obtain a new Julia set homeomorphic to a Julia set of a rational map; the dynamics of the two polynomials are reflected in the dynamics of the mated rational map. Here, I investigate the Julia sets of self-matings of generalized starlike quadratic polynomials, which enjoy relatively simple combinatorics. The points in the Julia sets of the mated rational maps are completely classified according to their topology. The presence and location of buried points in these Julia sets are addressed. The interconnections between complex dynamics, combinatorics, symbolic dynamics and Thurston's lamination theory are explored and utilized. The results are then extended to "quasi-self-matings".

• ### Aspects of Supercompactness, HOD and Set Theoretic Geology

Author:
Shoshana Friedman
Year of Dissertation:
2009
Program:
Mathematics
Arthur Apter
Abstract:

In this thesis, we study HOD, primarily in the context of large cardinals and GCH. Chapter 1 contains our introductory comments and preliminary remarks. In Chapter 2, we extend a property of HOD-supercompactness due to Sargsyan to various models of set theory containing supercompact cardinals. In doing so, we develop a new method for coding sets while preserving GCH. In Chapter 3, we extend this alternative method of coding. This allows us to produce models of V=HOD and GCH in the presence of large cardinals (including supercompact cardinals). In the remaining chapters, we use this coding to extend a variety of earlier results. In Chapter 4, we generalize theorems about the Ground Axiom to models with supercompact cardinals that satisfy GCH. In Chapter 5, we extend results in set theoretic geology to models that satisfy GCH. Finally, in Chapter 6, we use the coding to produce a model of the Wholeness Axiom, V=HOD and GCH.

• ### The Witt Ring of a Smooth Curve with Good Reduction over a Local Field

Author:
Jeanne Funk
Year of Dissertation:
2012
Program:
Mathematics
Raymond Hoobler
Abstract:

The modern study of bilinear forms has a rich history beginning with Witt's work over fields in the 1930's, when he defined a ring structure on the set of anisotropic forms over a field. It was revived, notably by Pfister, in the 1960's. With the advent of algebraic K-Theory, much of the theory of quadratic forms over fields was generalized to a theory of quadratic spaces over rings. In the 1960's and 1970's Knebusch, among others, formulated a compatible theory for quadratic forms over schemes in which a ring analogous to Witt's ring of anisotropic forms is prominent. Calculation of such "Witt rings" is a problem of interest in modern algebraic geometry. This thesis focuses on the calculation of the Witt ring of a smooth geometrically connected curve with good reduction over a local field. As a sub-problem, we calculate the Witt ring of a smooth gemetrically connected curve over a finite field. We present a generalization to the category of sheaves of the filtration of the Witt ring by powers of its fuundamental ideal of even rank elements. This yields a filtration by global sections which we study using \'{e}tale cohomology. In the cases of interest here, this allows us to describe the Witt classes of a curve in terms of the classical invarients rank, signed discriminant, and Witt invariant.

• ### Asymptotics for the parabolic, hyperbolic, and elliptic Eisenstein series through hyperbolic and elliptic degeneration

Author:
Daniel Garbin
Year of Dissertation:
2009
Program:
Mathematics
Jay Jorgenson
Abstract:

Let $\Gamma$ be a Fuchsian group of the first kind acting on the hyperbolic upper half plane $\mathbb{H}$, and let $M = \Gamma \backslash \mathbb{H}$ be the associated (connected) finite volume hyperbolic Riemann surface. We will allow the presence of both parabolic and elliptic elements as part of the group $\Gamma$, so that the surface has cusps (coming from parabolic elements) and conical points (coming from elliptic elements). To each primitive $\Gamma$-inconjugate parabolic element there is an associated parabolic Eisentein series which is more commonly referred to in the literature as the non-holomorphic Eisenstein series. If $\gamma$ is a primitive $\Gamma$-inconjugate hyperbolic element, then following the work due to Kudla and Millson, there is an associated hyperbolic Eisenstein series. More recently, Jorgenson and Kramer have introduced an elliptic Eisenstein series associated to a primitive $\Gamma$-inconjugate elliptic element $\gamma$ of the discontinous group $\Gamma$. \hskip 0.2in In this note, we look at the behavior of these Eisenstein series on families of hyperbolic Riemann surfaces of finite volume. In particular, there are two types of families that we study. The first family is obtained by hyperbolic degeneration which is a process that involves pinching primitive simple closed geodesic. The second family is obtained by elliptic degeneration, a process in which the order of ramification becomes unbounded, namely the order of elliptic fixed points associated to the conical points of the surface runs off to infinity. The main results are as follows. The Eisenstein series that are not associated to degenerating elements will converge to their correspondents in the limiting surface. For the Eisenstein series that are associated to degenerating elements the situation is as follows. In the case of hyperbolic degeneration, the hyperbolic Eisenstein series associated to a pinched geodesic will converge (up to a multiplicative factor) to the parabolic Eisenstein series associated to the newly developed cusp(s) in the limit surface. In the case of elliptic degeneration, a strikingly similar result occurs since the elliptic Eisenstein series associated to a degenerating conical point converges (up to a multiplicative factor) to the parabolic Eisenstein series associated to the newly developed cusp in the limit surface. The striking similarity lays in the fact that the above multiplicative factors involve the parameters defining the two type of degeneration, namely the length of the pinched geodesic in the case of hyperbolic degeneration and the angle of the pinched cone in the case of elliptic degeneration.

• ### Divisible Groups in the K-theory Completion of SU(n)

Author:
Peter Gregory
Year of Dissertation:
2011
Program:
Mathematics
Robert Thompson
Abstract:

I use the results of Bendersky and Thompson for the E(1)-based E2 -term of S2n+1, K E2s,tS{2n+1}, and the results of Bendersky and Davis concerning the v1-periodic groups of SU(n) to compute the E(1)-based E2-term for X=SU(n) for all primes p. This computation is performed using the Bendersky Thompson spectral sequence for SU(n). For spaces like SU(n) this spectral sequence converges to homotopy groups of the K-theory completion of SU(n). Of particular interest is the existence of infinitely many divisible groups in the homotopy groups of the K-theory completion of SU which offers an example of how E-completion does not commute with direct limits.

• ### Divisible Groups in the K-theory Completion of SU(n)

Author:
Peter Gregory
Year of Dissertation:
2011
Program:
Mathematics
Robert Thompson
Abstract:

I use the results of Bendersky and Thompson for the E(1)-based E2 -term of S2n+1, K E2s,tS{2n+1}, and the results of Bendersky and Davis concerning the v1-periodic groups of SU(n) to compute the E(1)-based E2-term for X=SU(n) for all primes p. This computation is performed using the Bendersky Thompson spectral sequence for SU(n). For spaces like SU(n) this spectral sequence converges to homotopy groups of the K-theory completion of SU(n). Of particular interest is the existence of infinitely many divisible groups in the homotopy groups of the K-theory completion of SU which offers an example of how E-completion does not commute with direct limits.

• ### Groups, Complexity, Cryptology

Author:
Maggie Habeeb
Year of Dissertation:
2012
Program:
Mathematics
Delaram Kahrobaei
Abstract:

The field of non-commutative group based cryptography has flourished in the past twelve years with the increasing need for secure public key cryptographic protocols. This has led to an active line of research called non-abelian group based cryptography. In this work, I in collaboration with Delaram Kahrobaei and Vladimir Shpilrain introduce a new public key exchange protocol based on a group theoretic problem and propose an appropriate platform group for this protocol. This work can be found in \cite{HKS0} and \cite{HKS}. In addition, I in collaboration with Delaram Kahrobaei and Vladimir Shpilrain propose two new secret sharing schemes that utilize non-abelian groups. These schemes have some advantages over Shamir's secret sharing scheme (see \cite{HKS2} for the full paper). We propose a class of groups, namely small cancellation groups, to implement these secret sharing schemes. Choosing the platform groups used in group based cryptographic protocols is vital to their security. D. Kahrobaei and B. Eick proposed in \cite{EK04} polycyclic groups as a potential platform for these cryptographic protocols. Polycyclic groups were also proposed as platform groups for group based cryptographic protocols in \cite{khan} and \cite{KA09}. An important feature of polycyclic groups, and hence finitely generated nilpotent groups, is that they are linear. I in collaboration with Delaram Kahrobaei considered the complexity of an embedding of a finitely generated torsion free nilpotent group into a linear group (see \cite{HK}). We determined the complexity of an algorithm introduced by W. Nickel in \cite{nickel} that determined a $\mathbb{Q}$-basis for a finite dimensional faithful $G$-module, which gives a bound on the dimension of the matrices produced. In \cite{HK} we also modified Nickel's algorithm for building a $\mathbb{Q}$-basis in order to improve the running time of the algorithm.

• ### Derived noncommutative deformation theory

Author:
Joseph Hirsh
Year of Dissertation:
2013
Program:
Mathematics
John Terilla
Abstract:

We define derived deformation theory with parameters over an operad $O$, and prove that the $\infty$-category of such theories is equivalent to the $\infty$-category of $O^{!}$-algebras.

• ### Martingales for Uniformly Quasisymmetric Circle Endomorphisms

Author:
Yunchun Hu
Year of Dissertation:
2014
Program:
Mathematics
Yunping Jiang
Abstract:

The main subject studied in this thesis is the space of all uniformly quasisymmetric circle endomorphisms preserving the Lebesgue measure. Although many of our arguments work for any degree d≥2, our proof will be mainly written for degree 2 maps. We will introduce a sequence of Markov partitions of the unit circle by using preimages of the fixed point of such circle endomorphism f. The uniform quasisymmetry condition is equivalent to the bounded nearby geometry condition of the Markov partitions. In Chapter 2 of this thesis, for each f, we use the Lebesgue invariant condition and the bounded geometry property to construct a martingale sequence {Xf,k} which has a L1 limiting function Xf on the dual symbolic space. We also show that the limiting martingale is invariant under symmetric conjugacy. The classical Hilbert transform introduces an almost complex structure on the space of all uniformly quasisymmetric circle endomorphisms that preserve the Lebesgue measure. This is presented in Chapter 3. In Chapters 4 & 5, we study locally constant limiting martingales and the related rigidity problems. A locally constant limiting martingale is the limit of a martingale sequence {Xk} of length n for some n≥0, i.e. the limiting martingale X=Xk for some n. We prove the rigidity problem for martingale sequence of length n≤4. That is, there is a unique way to construct a sequence of Markov partitions if the given limiting martingale Xf is equal to Xf,n for some n≤4. One of the consequences is that if two martingale sequences {Xf,k} and {Xg,k} have the same limit and both have length n≤4, where f and g are two uniformly quasisymmetric circle endomorphisms preserving the Lebesgue measure, then f=g. Another consequence is that if {Xf,k} has length n≤4, then there is no other map in the symmetric conjugacy class of f that preserves Lebesgue measure. In the class of uniformly symmetric circle endomorphisms, we prove that q(z)=z2, which has martingale sequence {Xq,k=2} for any k, is the only map whose limiting martingale is locally constant. Finally, we construct an analytic expanding circle endomorphism which preserves the Lebesgue measure and is a quasisymmetirc conjugate of q(z)=z2, i.e. f=hqh-1. We show that the conjugacy h is symmetric at one point but not symmetric on the whole unit circle.

• ### The Asymptotic Dirichlet Problems on manifolds with unbounded negative curvature

Author:
Ran Ji
Year of Dissertation:
2014
Program:
Mathematics
Jozef Dodziuk
Abstract:

Elton P. Hsu used probabilistic method to show that the asymptotic Dirichlet problem is uniquely solvable if the curvature satisfies the condition $-C e^{(2-\eta)r(x)} \leq K_M(x)\leq -1$ with $\eta>0$. We give an analytical proof of the same statement. In addition, using this new approach we are able to establish two boundary Harnack inequalities under the curvature condition $-C e^{(2/3-\eta)r(x)} \leq K_M(x)\leq -1$ with $\eta>0$. This implies that there is a natural homeomorphism between the Martin boundary and the geometric boundary of $M$. As far as we know, this is the first result of this kind under unbounded curvature conditions. Our proofs are modifications of arguments due to M. T. Anderson and R. Schoen.