# Alumni Dissertations and Theses

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### 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.

### Martingales for Uniformly Quasisymmetric Circle Endomorphisms

Author:Yunchun HuYear of Dissertation:2014Program:MathematicsAdvisor:Yunping JiangAbstract: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 endomorphismf . The uniform quasisymmetry condition is equivalent to the bounded nearby geometry condition of the Markov partitions. In Chapter 2 of this thesis, for eachf , we use the Lebesgue invariant condition and the bounded geometry property to construct a martingale sequence {X } which has a_{f,k}L 1 limiting functionX 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 {_{f}X } of length_{k}n for somen ≥0, i.e. the limiting martingaleX=X for some_{k}n . We prove the rigidity problem for martingale sequence of lengthn ≤4. That is, there is a unique way to construct a sequence of Markov partitions if the given limiting martingaleX is equal to_{f}X for some_{f,n}n ≤4. One of the consequences is that if two martingale sequences {X } and {_{f,k}X } have the same limit and both have length_{g,k}n ≤4, wheref andg are two uniformly quasisymmetric circle endomorphisms preserving the Lebesgue measure, thenf=g . Another consequence is that if {X } has length_{f,k}n ≤4, then there is no other map in the symmetric conjugacy class off that preserves Lebesgue measure. In the class of uniformly symmetric circle endomorphisms, we prove thatq(z)=z , which has martingale sequence {2 X =2} for any_{q,k}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 ofq(z)=z , i.e.2 f=hqh . We show that the conjugacy-1 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 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.

### Smooth Convergence Away From Singular Sets and Intrinsic Flat Continuity of Ricci Flow

Author:Sajjad LakzianYear of Dissertation:2013Program:MathematicsAdvisor:Chrisitina SormaniAbstract:In this thesis we provide a framework for studying the smooth limits of Riemannian metrics away from singular sets. We also provide applications to the non-degenrate neckpinch singularities in Ricci flow. We prove that if a family of metrics, $g_i$, on a compact Riemannian manifold, $M^n$, have a uniform lower Ricci curvature bound and converge to $g_\infty$ smoothly away from a singular set, $S$, with Hausdorff measure, $H^{n-1}(S) = 0$, and if there exists connected precompact exhaustion, $W_j$, of $M^n \setminus S$ satisfying $\diam_{g_i}(M^n) \le D_0 $, $\vol_{g_i}(\partial W_j) \le A_0 $ and $\vol_{g_i}(M^n \setminus W_j) \le V_j where \lim_{j\to\infty}V_j=0 $ then the Gromov-Hausdorff limit exists and agrees with the metric completion of $(M^n \setminus S, g_\infty)$. This is a strong improvement over prior work of the author with Sormani that had the additional assumption that the singular set had to be a smooth submanifold of codimension two. We have a second main theorem in which the Hausdorff measure condition on $S$ is replaced by diameter estimates on the connected components of the boundary of the exhaustion, $\partial W_j$. This second theorem allows for singular sets which are open subregions of the manifold. In addition, we show that the uniform lower Ricci curvature bounds in these theorems can be replaced by the existence of a uniform linear contractibility function. If this condition is removed altogether, then we prove that $\lim_{j\to \infty} d_{\mathcal{F}}(M_j', N')=0$, in which $M_j'$ and $N'$ are the settled completions of $(M, g_j)$ and $(M_\infty\setminus S, g_\infty)$ respectively and $d_{\mathcal{F}}$ is the Sormani-Wenger Intrinsic Flat distance. We present examples demonstrating the necessity of many of the hypotheses in our theorems. In the second part of this thesis, we study the Angenent-Caputo-Knopf's Ricci Flow through neckpinch singularities. We will explain how one can see the A-C-K's Ricci flow through a neckpinch singularity as a flow of integral current spaces. We then prove the continuity of this weak flow with respect to the Sormani-Wenger Intrinsic Flat (SWIF) distance.