# Alumni Dissertations

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### 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.

### Exploring platform (semi)groups for non-commutative key-exchange protocols

Author:Ha LamYear of Dissertation:2014Program:MathematicsAdvisor:Delaram KahrobaeiAbstract:In this work, my advisor Delaram Kahrobaei, our collaborator David Garber, and I explore polycyclic groups generated from number fields as platform for the AAG key-exchange protocol. This is done by implementing four different variations of the length-based attack, one of the major attacks for AAG, and submitting polycyclic groups to all four variations with a variety of tests. We note that this is the first time all four variations of the length-based attack are compared side by side. We conclude that high Hirsch length polycyclic groups generated from number fields are suitable for the AAG key-exchange protocol. Delaram Kahrobaei and I also carry out a similar strategy with the Heisenberg groups, testing them as platform for AAG with the length-based attack. We conclude that the Heisenberg groups, with the right parameters are resistant against the length-based attack. Another work in collaboration with Delaram Kahrobaei and Vladimir Shpilrain is to propose a new platform semigroup for the HKKS key-exchange protocol, that of matrices over a Galois field. We discuss the security of HKKS under this platform and advantages in computation cost. Our implementation of the HKKS key-exchange protocol with matrices over a Galois field yields fast run time.

### Combinatorial Properties of Polyiamonds

Author:Christopher LarsonYear of Dissertation:2014Program:MathematicsAdvisor:Joseph MalkevitchAbstract:Polyiamonds are plane geometric figures constructed by pasting together equilateral triangles edge-to-edge. It is shown that a diophantine equation involving vertices of degrees 2, 3, 5 and 6 holds for all polyiamonds; then an Eberhard-type theorem is proved, showing that any four-tuple of non-negative integers that satisfies the diophantine equation can be realized geometrically by a polyiamond. Further combinatorial and graph-theoretic aspects of polyiamonds are discussed, including a characterization of those polyiamonds that are three-connected and so three-polytopal, a result on Hamiltonicity, and constructions that use minimal numbers of triangles in realizing four-vectors.

### The Geometry of Lattice-Gauge-Orbit Space

Author:Michael LauferYear of Dissertation:2011Program:MathematicsAdvisor:Peter OrlandAbstract:In this paper, the Riemannian geometry of gauge-orbit space on the lattice with open boundary conditions is explored. It is shown how the metric and inverse metric tensors can be calculated, and further how the Ricci curvature might be calculated. The metric tensor and the inverse metric tensor are calculated for special cases, and some conjectures about the curvature of the space are made, which, if true, would move towards implying a mass gap in the theory.

### Resplendent models generated by indiscernibles

Author:Whanki LeeYear of Dissertation:2013Program:MathematicsAdvisor:Roman KossakAbstract:In this thesis I study the question: Which first-order structures are generated by indiscernibles? J. Schmerl showed that if

L<\italic> is a finite language, every countable recursively saturated L<\italic>-structure in which a form of coding of finite functions is available is generated by indiscernibles. Further, he showed that such a structure has arbitrarily large extensions which are generated by a set of indiscernibles, resplendent, and L<\italic>_{&infin,&omega}-equivalent to the original structure. Proofs of these theorems are complex and use a combinatorial lemma whose proof in Schmerl's paper has an acknowledged gap. I offer a complete proof of a more direct combinatorial lemma from which Schmerl's theorems follow. The other subject of this thesis is cofinal extensions of linearly ordered structures. It is related to the work of R. Kaye who used a weak notion of saturation to give a sufficient condition under which a countable model of PA -<\super> has a proper elementary cofinal extension. I give two different proofs of the fact that every countable recursively saturated linearly ordered structure with no last element has a proper cofinal elementary extension. ### Dual Graphs and PoincarĂ© Series of Valuations

Author:Charles LiYear of Dissertation:2012Program:MathematicsAdvisor:Hans SchoutensAbstract:Valuations on function fields of dimension two have been studied from the perspectives of dual graphs, generating sequences, Poincare series, and the valuative tree, among others. The goal of this dissertation is to greater unify these various approaches. Spivakovsky's dual graphs are used to calculate the Poincare series of non-divisorial valuations. With Galindo's results in the divisorial case already known, the equivalence of Poincare series with dual graphs is shown. A new elementary constructive proof of minimal generating sequences for non-divisorial valuations is given along the way, using only modest prerequisites from number theory. It is fair to say that the proof of minimal generating sequences is the crux of this dissertation, while the results on Poincare series are all corollaries.

### Problems in additive number theory

Author:Zeljka LjujicYear of Dissertation:2011Program:MathematicsAdvisor:Melvyn NathansonAbstract:In the first chapter we obtain the Biro-type upper bound for the smallest period of B in the case when A is a finite multiset of integers and B is a multiset such that A and B are t-complementing multisets of integers. In the second chapter we answer an inverse problem for lattice points proving that if K is a compact subset of R×R such that K+Z×Z=R×R then the integer points of the difference set of K is not contained on the coordinate axes, Z×{0}U{0}×Z. In the third chapter we show that there exist infinite sets A and M of positive integers whose partition function has weakly superpolynomial but not superpolynomial growth. The last chapter deals with the size of a sum of dilates 2·A+k·A. We prove that if k is a power of an odd prime or product of two primes and A a finite set of integers such that |A|>8k^k, then |2· A+k·A|≥ (k+2)|A|-k^2-k+2.

### Total Variation of Gaussian Processes and Local Times of Associated Levy Processes

Author:Jonathan LovellYear of Dissertation:2009Program:MathematicsAdvisor:Michael MarcusAbstract:Results of Taylor and Marcus and Rosen on the total variation of Gaussian processes and local times of associated symmetric stable processes are extended to a large class of symmetric Lévy processes. In this extension, the increments variance of the Gaussian process is generalized to a regularly varying function with index 0<α< 2. The result also includes a generalization of the total variation function.