# Alumni Dissertations and Theses

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### 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.

### The Margulis Region in Hyperbolic 4-space

Author:Viveka ErlandssonYear of Dissertation:2013Program:MathematicsAdvisor:Ara BasmajianAbstract:Given a discrete subgroup G of the isometries of n-dimensional hyperbolic space there is always a region kept precisely invariant under the stabilizer of a parabolic fixed point, called the Margulis region. This region corresponds to thin pieces in Thurston's thick-thin decomposition of the quotient manifold (or orbifold) M = H

n /G. In particular, the components of the Margulis region given by parabolic fixed points are related to the cusps of M. In dimensions 2 and 3 the Margulis region and the corresponding cusps are well-understood. In these dimensions parabolic isometries are conjugate to Euclidean translations and it follows that the Margulis region corresponding to a parabolic fixed point in dimensions 2 and 3 is always a horoball. In higher dimensions the region has in general a more complicated shape. This is due to the fact that parabolic isometries in dimensions 4 and higher can have a rotational part, which are called screw parabolic elements. There are examples due to Ohtake and Apanasov of discrete groups containing screw parabolic elements for which there is no precisely invariant horoball. Hence the corresponding Margulis region cannot be a horoball. It is natural to wonder about the shape of the Margulis region corresponding to a screw parabolic fixed point, and how it differs from that of a horoball. We describe the asymptotic behavior of the boundary of the Margulis region in hyperbolic 4-space corresponding to the fixed point of a screw parabolic isometry with an irrational rotation of bounded type. As a corollary we show that the region is quasi-isometric to a horoball. That is, there is a quasi-isometry of hyperbolic 4-space that maps the Margulis region to a horoball. Although it is known that two screw parabolic isometries with distinct irrational rotational parts are not conjugate by any quasi-isometry of H4 , this corollary implies that their corresponding Margulis regions (in the bounded type case) are quasi-isometric.### The Length Spectrum Metric on the Teichmuller Space of a Flute Surface

Author:Ozgur EvrenYear of Dissertation:2013Program:MathematicsAdvisor:Ara BasmajianAbstract:The topology defined by the length spectrum metric on the Teichmuller space of an infinite type surface, in contrast to finite type surfaces, need not be the same as the topology defined by the Teichmuller metric. In this thesis, we study the equivalence of these topologies on a particular kind of infinite type surface, called the flute surface. Following a construction by Shiga and using additional hyperbolic geometric estimates, we obtain sufficient conditions in terms of length parameters for these two metrics to be topologically inequivalent. Next, we construct infinite parameter families of quasiconformally distinct flute surfaces, both with fixed and varying boundary data, with the property that the length spectrum metric is not topologically equivalent to the Teichmuller metric.