# Alumni Dissertations

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### 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.

### Cohomological aspects of complete reducibility of representations

Author:Ioannis FarmakisYear of Dissertation:2009Program:MathematicsAdvisor:Martin MoskowitzAbstract:yes

### On the Dynamics of Quasi-Self-Matings of Generalized Starlike Complex Quadratics and the Structure of the Mated Julia Sets

Author:Ross FlekYear of Dissertation:2009Program:MathematicsAdvisor:Linda KeenAbstract:It has been shown that, in many cases, Julia sets of complex polynomials can be "glued" together to obtain a new Julia set homeomorphic to a Julia set of a rational map; the dynamics of the two polynomials are reflected in the dynamics of the mated rational map. Here, I investigate the Julia sets of self-matings of generalized starlike quadratic polynomials, which enjoy relatively simple combinatorics. The points in the Julia sets of the mated rational maps are completely classified according to their topology. The presence and location of buried points in these Julia sets are addressed. The interconnections between complex dynamics, combinatorics, symbolic dynamics and Thurston's lamination theory are explored and utilized. The results are then extended to "quasi-self-matings".

### Aspects of Supercompactness, HOD and Set Theoretic Geology

Author:Shoshana FriedmanYear of Dissertation:2009Program:MathematicsAdvisor:Arthur ApterAbstract:In this thesis, we study HOD, primarily in the context of large cardinals and GCH. Chapter 1 contains our introductory comments and preliminary remarks. In Chapter 2, we extend a property of HOD-supercompactness due to Sargsyan to various models of set theory containing supercompact cardinals. In doing so, we develop a new method for coding sets while preserving GCH. In Chapter 3, we extend this alternative method of coding. This allows us to produce models of V=HOD and GCH in the presence of large cardinals (including supercompact cardinals). In the remaining chapters, we use this coding to extend a variety of earlier results. In Chapter 4, we generalize theorems about the Ground Axiom to models with supercompact cardinals that satisfy GCH. In Chapter 5, we extend results in set theoretic geology to models that satisfy GCH. Finally, in Chapter 6, we use the coding to produce a model of the Wholeness Axiom, V=HOD and GCH.

### The Witt Ring of a Smooth Curve with Good Reduction over a Local Field

Author:Jeanne FunkYear of Dissertation:2012Program:MathematicsAdvisor:Raymond HooblerAbstract:The modern study of bilinear forms has a rich history beginning with Witt's work over fields in the 1930's, when he defined a ring structure on the set of anisotropic forms over a field. It was revived, notably by Pfister, in the 1960's. With the advent of algebraic K-Theory, much of the theory of quadratic forms over fields was generalized to a theory of quadratic spaces over rings. In the 1960's and 1970's Knebusch, among others, formulated a compatible theory for quadratic forms over schemes in which a ring analogous to Witt's ring of anisotropic forms is prominent. Calculation of such "Witt rings" is a problem of interest in modern algebraic geometry. This thesis focuses on the calculation of the Witt ring of a smooth geometrically connected curve with good reduction over a local field. As a sub-problem, we calculate the Witt ring of a smooth gemetrically connected curve over a finite field. We present a generalization to the category of sheaves of the filtration of the Witt ring by powers of its fuundamental ideal of even rank elements. This yields a filtration by global sections which we study using \'{e}tale cohomology. In the cases of interest here, this allows us to describe the Witt classes of a curve in terms of the classical invarients rank, signed discriminant, and Witt invariant.