# Alumni Dissertations

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### 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.

### 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.

### The Hilbert Projective Metric, Multi-type Branching Processes and Mathematical Biology: a Model of the Evolution of Resistance

Author:Christopher McCarthyYear of Dissertation:2010Program:MathematicsAdvisor:Yunping JiangAbstract:Bacteria, viruses, or cancer cells, by means of mutation and replication, are sometimes able to escape the selective pressure exerted by treatment. This is called the development, or evolution, of resistance. This dissertation is a study of some of the mathematics underlying a model of resistance put forth by Iwasa, Michor, and Nowak (IMN) "Evolutionary Dynamics of Invasion and Escape" (2003, 2004). In the IMN model the pre-treatment phase is modeled as a determinist dynamical system using Eigen and Schuster's quasispecies theory of evolution. It is assumed that at the start of treatment the system has reached an invariant distribution: the quasispecies equilibrium eigenvector. The equations of the quasispecies theory can be viewed as projections of linear differential equations onto hyperplanes and their asymptotic behavior can be understood via Birkhoff's Projective Contraction Theorem (1957), which is related to the Perron-Frobenius Theorem. An understanding of Birkhoff's contraction theorem requires an understanding of the Hilbert Projective Metric and so we develop an extensive collection of useful related results, some novel, about cones, hyperplanes, and the Hilbert Metric. In the IMN model, the post-treatment phase is modeled as a stochastic multi-type branching process on the various mutant types. The key calculation is the vector of extinction probabilities: the i entry of the vector being the probability that a process, starting with a single mutant of type i, will eventually go extinct (under the selective pressure of treatment). The techniques for calculating these extinction probabilities involve the use of multi-type probability generating functions (PGF's). We prove results about the existence of continuous multi-type PGF's and branching processes. Our proofs involve customizing techniques from the theory of differential equations in complex vector spaces, and then applying results from the theory of several complex variables. We also develop a method to numerically calculate the vector of extinction probabilities. The pre and post-treatment models are fitted together and the probability of a successful treatment is numerically calculated using a combination of standard techniques from numerical analysis together with insights gained from our examination of the mathematical aspects of the model. Our investigation leads to a phenomena somewhat reminiscent of Eigen's error catastrophe theory. Supplementary materials: hyper-linked PDF of dissertation, Matlab m-files.

### Asymptotic Invariants and Flatness of Local Endomorphisms

Author:Nikita MiasnikovYear of Dissertation:2014Program:MathematicsAdvisor:Lucien SzpiroAbstract:For a local endomorphism of a noetherian local ring we introduce 3 asymptotic invariants one of which we call entropy. We use this notion of entropy to extend numerical conditions in Kunz' regularity criterion to every contracting endomorphism of a noetherian local ring, and to give a characteristic-free interpretation of the definition of Hilbert-Kunz multiplicity. We also show that every finite endomorphism of a complete noetherian local ring of equal characteristic can be lifted to a finite endomorphism of a complete regular local ring. The local ring of an algebraic or analytic variety at a point fixed by a finite self-morphism inherits a local endomorphism whose entropy is well-defined. This situation arises at the vertex of the affine cone over a projective variety with a polarized self-morphism, where we compare entropy with degree.

### Algebraic Models for the Free Loop Space and Differential Forms of a Manifold

Author:Micah MillerYear of Dissertation:2011Program:MathematicsAdvisor:Mahmoud ZeinalianAbstract:Our initial goal is to give a chain level description of the string topology loop product for a large class of spaces. This effort is described in two parts; the first uses Brown's theory of twisting cochains to obtain a model for the free loop space of a manifold and the second constructs a minimal model for the Frobenius algebra of differential forms of a manifold. The first part defines the loop product for closed, oriented manifolds and Poincare Duality spaces. The second part is an attempt to understand the minimal model for the Frobenius algebra of a manifold, with the idea of extending the methods in the first section to define the loop product for open manifolds. Brown's theory of twisting cochains provides a chain model of a principal G-bundle and its associated bundles. The free loop space is obtained by considering the path space fibration, and taking the associated bundle with the based loop space acting on itself by conjugation. Given a twisting cochain, then, we obtain a chain model of LM using Brown's theory. To describe the chain-level loop product in this setting, we need a model for the intersection product in the chains on M. For this, we use the cyclic commutative infinity algebra structure on the homology of M. Such a description would give a chain level description of the string topology loop product for open manifolds. Instead of using the cyclic commutative algebra, we could have used the Frobenius algebra structure. One would expect that the Frobenius infinity algebra can be used to show the necessary relations to define the loop product. Then given the Frobenius infinity algebra on the homology of M for an open manifold, we would have a chain level description of the loop product. The purpose of Section 3 is to gain a better understanding of the Frobenius infinity algebra on the cohomology of M. The Frobenius algebra, induced by the wedge product and Poincare Duality, is well understood; the structure on the level of forms inducing the Frobenius algebra is less well understood. We use the language of operads, dioperads, and properads and Koszul duality to give a definition of Frobenius infinity algebra. We also use descriptions of the transfer of structure using trees and integrating over cells in the moduli space of metrised ribbon graphs. When M is closed and oriented, these tools allow us to build a minimal model for the Frobenius algebra of differential forms on M and to compare it with the cyclic commutative infinity algebra.