# Alumni Dissertations and Theses

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### Some applications of noncommutative groups and semigroups to information security

Author:Lisa BrombergYear of Dissertation:2015Program:MathematicsAdvisor:Vladimir ShpilrainAbstract:We present evidence why the Burnside groups of exponent 3 could be a good candidate for a platform group for the HKKS semidirect product key exchange protocol. We also explore hashing with matrices over SL_2(F_p), and compute bounds on the girth of the Cayley graph of the subgroup of SL_2(F_p) for specific generators A,B. We demonstrate that even without optimization, these hashes have comparable performance to hashes in the SHA family.

### Late Points of Projections of Planar Symmetric Random Walks on the Lattice Torus

Author:Michael CarlisleYear of Dissertation:2012Program:MathematicsAdvisor:Jay RosenAbstract:We examine the

cover time and set oflate points of a symmetric random walk on Z2 projected onto the torus Z2 _{K}. This extends the work done for the simple random walk in [Late Points, DPRZ, 2006] to a large class of random walks. The approach uses comparisons between planar and toral hitting times and distributions on annuli, and uses only random walk methods. There are also generalizations of Green's functions, hitting times, and hitting distributions on Z2 and Z2 _{K}which are of independent interest.### Force to change large cardinal strength

Author:Erin CarmodyYear of Dissertation:2015Program:MathematicsAdvisor:Joel HamkinsAbstract:This dissertation includes many theorems which show how to change large cardinal properties with forcing. I consider in detail the degrees of inaccessible cardinals (an analogue of the classical degrees of Mahlo cardinals) and provide new large cardinal definitions for degrees of inaccessible cardinals extending the hyper-inaccessible hierarchy. I showed that for every cardinal $\kappa$, and ordinal $\alpha$, there is a notion of forcing $\mathbb{P}$ such that $\kappa$ is still $\beta$-inaccessible in the extension, for every $\beta < \alpha$, but not $\alpha$-inaccessible. I also consider Mahlo cardinals and degrees of Mahlo cardinals. I showed that for every cardinal $\kappa$, and ordinal $\alpha$, there is a notion of forcing $\mathbb{P}$ such that for every $\beta < \alpha$, the cardinal $\kappa$ is still $\beta$-Mahlo in the extension, but not $\alpha$-Mahlo. I also show that a cardinal $\kappa$ which is Mahlo in the ground model can have every possible inaccessible degree in the forcing extension, but no longer be Mahlo there. The thesis includes a collection of results which give forcing notions which change large cardinal strength from weakly compact to weakly measurable, including some earlier work by others that fit this theme. I consider in detail measurable cardinals and Mitchell rank. I show how to change a class of measurable cardinals by forcing to an extension where all measurable cardinals above some fixed ordinal $\alpha$ have Mitchell rank below $\alpha.$ Finally, I consider supercompact cardinals, and strongly compact cardinals. I show how to change the Mitchell rank for supercompactness for a class of cardinals.

### Algorithmic properties of poly-Z groups and secret sharing using non-commutative groups

Author:Bren CavalloYear of Dissertation:2015Program:MathematicsAdvisor:Delaram KahrobaeiAbstract:Computational aspects of polycyclic groups have been used to study cryptography since 2004 when Eick and Kahrobaei proposed polycyclic groups as a platform for conjugacy based cryptographic protocols. In the first chapter we study the conjugacy problem in polycyclic groups and construct a family of torsion-free polycyclic groups where the uniform conjugacy problem over the entire family is at least as hard as the subset sum problem. We further show that the conjugacy problem in these groups is in NP, implying that the uniform conjugacy problem is NP-complete over these groups. This is joint work with Delaram Kahrobaei. We also present an algorithm for the conjugacy problem in groups of the form $\Z^n \rtimes_\phi \Z$. We continue by studying automorphisms of poly-$\Z$ groups and successive cyclic extensions of arbitrary groups. We study a certain kind of extension that we call ``deranged", and show that the automorphisms of the resulting group have a strict form. We also show that the automorphism group of a group obtained by iterated extensions of this type contains a non-abelian free group if and only if the original base group does. Finally we show that it is possible to verify that a finitely presented by infinite cyclic group is finitely presented by infinite cyclic, but that determining that a general finitely presented group is finitely generated by infinite cyclic is undecidable. We then discuss implications the latter result has for calculating the Bieri-Neumann-Strebel invariant. This is joint work with Jordi Delgado, Delaram Kahrobaei, Ha Lam, and Enric Ventura and is currently in preparation. In the final chapter we discuss secret sharing schemes and variations. We begin with classical secret sharing schemes and present variations that allow them to be more practical. We then present a secret sharing scheme due to Habeeb, Kahrobaei, and Shpilrain. Finally, we present an original adjustment to their scheme that involves the shortlex order on a group and allows less information to be transmitted each time a secret is shared. Additionally, we propose additional steps that allow participants to update their information independently so that the scheme remains secure over multiple rounds. This is joint work with Delaram Kahrobaei.

### Uniqueness Theorems for Some Nonlinear Parabolic Equations

Author:Yimao ChenYear of Dissertation:2012Program:MathematicsAdvisor:Leon KarpAbstract:We study the uniqueness of solutions of the Cauchy problem of two nonlinear parabolic equations in this thesis. We first study the uniqueness of the solutions of the initial value problem associated with the infinity-Laplacian operators. We prove the uniqueness of solutions of the Cauchy problem for the infinity-Laplacian heat equation in a class of functions with exponential growth. We also study the uniqueness of the solutions of the evolution associated with the minimal surface equation. We obtain a new uniqueness class of solutions of the Cauchy problem for the parabolic minimal surface equation, which is reminiscent of the classical results for the heat equation.

### Geometric Characterization and Dynamics of Holomorphic maps

Author:Tao ChenYear of Dissertation:2013Program:MathematicsAdvisor:Yunping JiangAbstract:We prove the existence of the canonical Thurston obstruction for sub-hyperbolic semi-rational branched coverings when they are obstructed. Then we geometrically characterize meromorphic maps with exactly two asymptotic values and no critical values. We finish with the proof of non-existence of the invariant line fields for a family of entire functions

### Involutions in Arithmetic Geometry

Author:Anbo ChenYear of Dissertation:2013Program:MathematicsAdvisor:Bruce JordanAbstract:We first study the integral representation $L$ of $G=\langle \sigma \rangle$, where $\sigma$ is an involution. When $L=H_1(X, \mathbb{Z})$ for some algebraic curve $X$, we determine the structure $L$ completely by the the intersection of $J_+$ and $J_-$, where $J_{\pm}$ are the subvarieties of the Jabocian $J$ of $X$. Then, we study the structure of $L=H_1(X, \mathbb{Z})$ as the integral representation of Klein 4 group $G=\langle \sigma, \tau \rangle$, where $\sigma$ and $\tau$ are two commuting involutions. Computations are also included in our work.

### Some Results on Large Cardinals and the Continuum Function

Author:Brent CodyYear of Dissertation:2012Program:MathematicsAdvisor:Joel HamkinsAbstract:I prove several new relative consistency results concerning large cardinals and the continuum function.

### The Differentiability of Renormalized Triple Intersection Local Times

Author:Subir DhamoonYear of Dissertation:2013Program:MathematicsAdvisor:Jay RosenAbstract:The evolution of the theory of triple intersection times over the past, approximately, two decades has centered primarily on two dimensional Brownian Motion and planar symmetric stable processes. The one dimensional cases have gone largely unstudied. In this thesis, we examine the differentiability of renormalized triple intersection local times for the two aforementioned Markov processes in

R 1 . In more detail, we prove that the single partial derivative with respect to each spatial variable exists and show that each partial derivative is, in fact, jointly continuous in both space and time variables. During the course of our analysis, we discover that these results hold for the class of symmetric stable process for which 3/2<β<2.### Homotopy Batalin-Vilkovisky algebras, trivializing circle actions, and moduli space

Author:Gabriel C. Drummond-ColeYear of Dissertation:2010Program:MathematicsAdvisor:John TerillaAbstract:This thesis comprises two main results, one topological, one algebraic. The topological result is that an action of the framed little disks operad and a trivialization of the circle action within it determine an action of the Deligne-Mumford compactification of the moduli space of genus zero curves. The algebraic result is a description of the structure of minimal homotopy Batalin-Vilkovisky algebras and the the theorem that in the case that the Batalin-Vilkovisky operator and its higher homotopies are trivial, the remaining algebraic structure is a minimal homotopy hypercommutative algebra. These results are related to one another because the algebraic structures involved are representations of the homology of, respectively, the framed little disks and the Deligne-Mumford compactification.