# Alumni Dissertations

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### 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.

### Derived noncommutative deformation theory

Author:Joseph HirshYear of Dissertation:2013Program:MathematicsAdvisor:John TerillaAbstract: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.

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

Author:Ran JiYear of Dissertation:2014Program:MathematicsAdvisor:Jozef DodziukAbstract: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.

### Length spectrum metric and modified length spectrum metric on Teichmüller spaces

Author:Francisco Jimenez LopezYear of Dissertation:2013Program:MathematicsAdvisor:Jun HuAbstract:The length spectrum function defines a metric on the reduced Teichmüller space of a Riemann surface which is topologically equivalent, but not metrically equivalent to the Teichmüller metric if the Riemann surface is of finite topological type. As the first part of this work, in the reduced Teichmüller space of a Riemann surface of finite topological type, we find two points moving towards the boundary of the space along two continuous curves, such that the Teichmüller distance between them approaches infinity while their length spectrum distance approaches zero. Unfortunately, the length spectrum function does not define a metric on the (unreduced) Teichmüller space of a Riemann surface with boundary. In the second part of this work, we introduce a modified length spectrum function that does define a metric on this space. We show that if two points are close with respect to the Teichmüller metric, then they are also close in the modified length spectrum metric. We also show that the converse is not true. Finally, we prove that the (unreduced) Teichmüller space of a Riemann surface of finite topological type with non-empty boundary is not complete under the modified length spectrum metric.

### On the rank of 2-primary part of Selmer group of certain elliptic curves

Author:KWANG HYUN KIMYear of Dissertation:2012Program:MathematicsAdvisor:Victor KolyvaginAbstract:Kolyvagin proved very remarkable results on Mordell-Weil groups and Shafarevich-Tate groups of certain elliptic curves when a given Heegner point P

_{K}has infinite order in his series of papers. He also extended his result to odd prime l-primary part of Selmer group of higher rank with the assumption of existence of non-trivial Kolyvagin system. In this thesis, we will follow his Euler system method and verify that his method also works to prove the result on the rank of 2-primary part of Selmer group of higher rank with Strong non-zero conjecture.### The Admissible Dual of SL(2) of the Dyadic Numbers

Author:Terence Kivran-SwaineYear of Dissertation:2011Program:MathematicsAdvisor:Carlos MorenoAbstract:The admissible dual of SL

_{2}(Q _{2}) is constructed uniformly, based on a method adapted from the the theory of cuspidal types of GL_{2}(F).### On critical poins for Gaussian vectors with infinitely divisible squares

Author:Hana KoganYear of Dissertation:2009Program:MathematicsAdvisor:Michael MarcusAbstract:This paper is concerned with necessary conditions for infinite divisibility of the Gaussian squares with non-zero means. A Gaussian vector G with zero mean is said to have a critical point α, such that 0≤α≤∞ if the square of (G+α) is infinitely divisible for all |β| ≤ α and is not infinitely divisible for all |β|≥ α. We derive upper bound for the critical point of a Gaussian n-dimentional vector via the asymptotic analysis of its Laplace Transform.

### Non-commutative cryptography: Diffie-Hellman and CCA secure cryptosystems using matrices over group rings and digital signatures

Author:Charalambos KoupparisYear of Dissertation:2012Program:MathematicsAdvisor:Delaram KahrobaeiAbstract:As computing speed has been following Moore's law without any inclination of tapering out, the need for ever more secure cryptographic protocols is becoming more and more relevant. During the past one and a half decades the field of non-commutative (or on-abelian) group based cryptography has seen a surge in interest. Through this work we will present the classical Diffie-Hellman public key exchange protocol (DH PKE) and discuss two important notions related to it, the Computational Diffie-Hellman assumption and the Decision Diffie-Hellman assumption. We then proceed to look at a new platform group based on matrices over group srings and present work done by myself in collaboration with Delaram Kahrobaei and Vladimir Shpilrain. We discuss the viability of the new platform group and point out its benefits. Additionally, I in collaboration with Delaram Kahrobaei and Vladimir Shpilrain propose to use the new platform group in the Cramer-Shoup cryptosystem. We demonstrate how one can implement the system using our platform and prove that the system is still CCA-2 secure. Finally, we discuss the notion of classical digital signatures following the work of Goldwasser and Bellare and Schnorr. We then discuss some non-commutative digital signatures including those proposed by Ko, Choi, Cho and Lee, Wang and Hu Anjaneyulu, Reddy and Reddy and Chaum and van Antwerpen. We conclude by presenting work done my myself in conjunction with Delaram Kahrobaei which discusses a new non-commutative digital signature. We propose using groups for which the Conjugacy Search Problem is hard, or any group which is secure against length based attacks, such as polycyclic groups, as the platform for this signature.