# Alumni Dissertations

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### Cohomological aspects of complete reducibility of representations

Author:Ioannis FarmakisYear of Dissertation:2009Program:MathematicsAdvisor:Martin MoskowitzAbstract:yes

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

Author:Ross FlekYear of Dissertation:2009Program:MathematicsAdvisor:Linda KeenAbstract: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".

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

Author:Ross FlekYear of Dissertation:2009Program:MathematicsAdvisor:Linda KeenAbstract: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 FriedmanYear of Dissertation:2009Program:MathematicsAdvisor:Arthur ApterAbstract: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.

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

Author:Shoshana FriedmanYear of Dissertation:2009Program:MathematicsAdvisor:Arthur ApterAbstract: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 FunkYear of Dissertation:2012Program:MathematicsAdvisor:Raymond HooblerAbstract: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 GarbinYear of Dissertation:2009Program:MathematicsAdvisor:Jay JorgensonAbstract: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.

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

Author:Daniel GarbinYear of Dissertation:2009Program:MathematicsAdvisor:Jay JorgensonAbstract: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 GregoryYear of Dissertation:2011Program:MathematicsAdvisor:Robert ThompsonAbstract:I use the results of Bendersky and Thompson for the

E(1) -basedE -term of_{2}S ,2n+1 , and the results of Bendersky and Davis concerning theK E_{2}s,t S{2n+1} v -periodic groups of_{1}SU(n) to compute theE(1) -basedE -term for_{2}X=SU(n) for all primesp . This computation is performed using the Bendersky Thompson spectral sequence forSU(n) . For spaces likeSU(n) this spectral sequence converges to homotopy groups of theK -theory completion ofSU(n) . Of particular interest is the existence of infinitely many divisible groups in the homotopy groups of theK -theory completion ofSU which offers an example of howE -completion does not commute with direct limits.### Groups, Complexity, Cryptology

Author:Maggie HabeebYear of Dissertation:2012Program:MathematicsAdvisor:Delaram KahrobaeiAbstract: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.