# Alumni Dissertations and Theses

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### 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.

### Asymptotics for the parabolic, hyperbolic, and elliptic Eisenstein series through hyperbolic and elliptic degeneration

Author:Daniel GarbinYear of Dissertation:2009Program:MathematicsAdvisor:Jay JorgensonAbstract:Let $\Gamma$ be a Fuchsian group of the first kind acting on the hyperbolic upper half plane $\mathbb{H}$, and let $M = \Gamma \backslash \mathbb{H}$ be the associated (connected) finite volume hyperbolic Riemann surface. We will allow the presence of both parabolic and elliptic elements as part of the group $\Gamma$, so that the surface has cusps (coming from parabolic elements) and conical points (coming from elliptic elements). To each primitive $\Gamma$-inconjugate parabolic element there is an associated parabolic Eisentein series which is more commonly referred to in the literature as the non-holomorphic Eisenstein series. If $\gamma$ is a primitive $\Gamma$-inconjugate hyperbolic element, then following the work due to Kudla and Millson, there is an associated hyperbolic Eisenstein series. More recently, Jorgenson and Kramer have introduced an elliptic Eisenstein series associated to a primitive $\Gamma$-inconjugate elliptic element $\gamma$ of the discontinous group $\Gamma$. \hskip 0.2in In this note, we look at the behavior of these Eisenstein series on families of hyperbolic Riemann surfaces of finite volume. In particular, there are two types of families that we study. The first family is obtained by hyperbolic degeneration which is a process that involves pinching primitive simple closed geodesic. The second family is obtained by elliptic degeneration, a process in which the order of ramification becomes unbounded, namely the order of elliptic fixed points associated to the conical points of the surface runs off to infinity. The main results are as follows. The Eisenstein series that are not associated to degenerating elements will converge to their correspondents in the limiting surface. For the Eisenstein series that are associated to degenerating elements the situation is as follows. In the case of hyperbolic degeneration, the hyperbolic Eisenstein series associated to a pinched geodesic will converge (up to a multiplicative factor) to the parabolic Eisenstein series associated to the newly developed cusp(s) in the limit surface. In the case of elliptic degeneration, a strikingly similar result occurs since the elliptic Eisenstein series associated to a degenerating conical point converges (up to a multiplicative factor) to the parabolic Eisenstein series associated to the newly developed cusp in the limit surface. The striking similarity lays in the fact that the above multiplicative factors involve the parameters defining the two type of degeneration, namely the length of the pinched geodesic in the case of hyperbolic degeneration and the angle of the pinched cone in the case of elliptic degeneration.

### Divisible Groups in the K-theory Completion of SU(n)

Author:Peter GregoryYear of Dissertation:2011Program:MathematicsAdvisor:Robert ThompsonAbstract:I use the results of Bendersky and Thompson for the

E(1) -basedE -term of_{2}S ,2n+1 , and the results of Bendersky and Davis concerning theK E_{2}s,t S{2n+1} v -periodic groups of_{1}SU(n) to compute theE(1) -basedE -term for_{2}X=SU(n) for all primesp . This computation is performed using the Bendersky Thompson spectral sequence forSU(n) . For spaces likeSU(n) this spectral sequence converges to homotopy groups of theK -theory completion ofSU(n) . Of particular interest is the existence of infinitely many divisible groups in the homotopy groups of theK -theory completion ofSU which offers an example of howE -completion does not commute with direct limits.### Groups, Complexity, Cryptology

Author:Maggie HabeebYear of Dissertation:2012Program:MathematicsAdvisor:Delaram KahrobaeiAbstract:The field of non-commutative group based cryptography has flourished in the past twelve years with the increasing need for secure public key cryptographic protocols. This has led to an active line of research called non-abelian group based cryptography. In this work, I in collaboration with Delaram Kahrobaei and Vladimir Shpilrain introduce a new public key exchange protocol based on a group theoretic problem and propose an appropriate platform group for this protocol. This work can be found in \cite{HKS0} and \cite{HKS}. In addition, I in collaboration with Delaram Kahrobaei and Vladimir Shpilrain propose two new secret sharing schemes that utilize non-abelian groups. These schemes have some advantages over Shamir's secret sharing scheme (see \cite{HKS2} for the full paper). We propose a class of groups, namely small cancellation groups, to implement these secret sharing schemes. Choosing the platform groups used in group based cryptographic protocols is vital to their security. D. Kahrobaei and B. Eick proposed in \cite{EK04} polycyclic groups as a potential platform for these cryptographic protocols. Polycyclic groups were also proposed as platform groups for group based cryptographic protocols in \cite{khan} and \cite{KA09}. An important feature of polycyclic groups, and hence finitely generated nilpotent groups, is that they are linear. I in collaboration with Delaram Kahrobaei considered the complexity of an embedding of a finitely generated torsion free nilpotent group into a linear group (see \cite{HK}). We determined the complexity of an algorithm introduced by W. Nickel in \cite{nickel} that determined a $\mathbb{Q}$-basis for a finite dimensional faithful $G$-module, which gives a bound on the dimension of the matrices produced. In \cite{HK} we also modified Nickel's algorithm for building a $\mathbb{Q}$-basis in order to improve the running time of the algorithm.

### Derived noncommutative deformation theory

Author:Joseph HirshYear of Dissertation:2013Program:MathematicsAdvisor:John TerillaAbstract:We define derived deformation theory with parameters over an operad $O$, and prove that the $\infty$-category of such theories is equivalent to the $\infty$-category of $O^{!}$-algebras.

### Martingales for Uniformly Quasisymmetric Circle Endomorphisms

Author:Yunchun HuYear of Dissertation:2014Program:MathematicsAdvisor:Yunping JiangAbstract:The main subject studied in this thesis is the space of all uniformly quasisymmetric circle endomorphisms preserving the Lebesgue measure. Although many of our arguments work for any degree

d ≥2, our proof will be mainly written for degree 2 maps. We will introduce a sequence of Markov partitions of the unit circle by using preimages of the fixed point of such circle endomorphismf . The uniform quasisymmetry condition is equivalent to the bounded nearby geometry condition of the Markov partitions. In Chapter 2 of this thesis, for eachf , we use the Lebesgue invariant condition and the bounded geometry property to construct a martingale sequence {X } which has a_{f,k}L 1 limiting functionX on the dual symbolic space. We also show that the limiting martingale is invariant under symmetric conjugacy. The classical Hilbert transform introduces an almost complex structure on the space of all uniformly quasisymmetric circle endomorphisms that preserve the Lebesgue measure. This is presented in Chapter 3. In Chapters 4 & 5, we study locally constant limiting martingales and the related rigidity problems. A locally constant limiting martingale is the limit of a martingale sequence {_{f}X } of length_{k}n for somen ≥0, i.e. the limiting martingaleX=X for some_{k}n . We prove the rigidity problem for martingale sequence of lengthn ≤4. That is, there is a unique way to construct a sequence of Markov partitions if the given limiting martingaleX is equal to_{f}X for some_{f,n}n ≤4. One of the consequences is that if two martingale sequences {X } and {_{f,k}X } have the same limit and both have length_{g,k}n ≤4, wheref andg are two uniformly quasisymmetric circle endomorphisms preserving the Lebesgue measure, thenf=g . Another consequence is that if {X } has length_{f,k}n ≤4, then there is no other map in the symmetric conjugacy class off that preserves Lebesgue measure. In the class of uniformly symmetric circle endomorphisms, we prove thatq(z)=z , which has martingale sequence {2 X =2} for any_{q,k}k , is the only map whose limiting martingale is locally constant. Finally, we construct an analytic expanding circle endomorphism which preserves the Lebesgue measure and is a quasisymmetirc conjugate ofq(z)=z , i.e.2 f=hqh . We show that the conjugacy-1 h is symmetric at one point but not symmetric on the whole unit circle.### The Asymptotic Dirichlet Problems on manifolds with unbounded negative curvature

Author:Ran JiYear of Dissertation:2014Program:MathematicsAdvisor:Jozef DodziukAbstract:Elton P. Hsu used probabilistic method to show that the asymptotic Dirichlet problem is uniquely solvable if the curvature satisfies the condition $-C e^{(2-\eta)r(x)} \leq K_M(x)\leq -1$ with $\eta>0$. We give an analytical proof of the same statement. In addition, using this new approach we are able to establish two boundary Harnack inequalities under the curvature condition $-C e^{(2/3-\eta)r(x)} \leq K_M(x)\leq -1$ with $\eta>0$. This implies that there is a natural homeomorphism between the Martin boundary and the geometric boundary of $M$. As far as we know, this is the first result of this kind under unbounded curvature conditions. Our proofs are modifications of arguments due to M. T. Anderson and R. Schoen.

### Length spectrum metric and modified length spectrum metric on Teichmüller spaces

Author:Francisco Jimenez LopezYear of Dissertation:2013Program:MathematicsAdvisor:Jun HuAbstract:The length spectrum function defines a metric on the reduced Teichmüller space of a Riemann surface which is topologically equivalent, but not metrically equivalent to the Teichmüller metric if the Riemann surface is of finite topological type. As the first part of this work, in the reduced Teichmüller space of a Riemann surface of finite topological type, we find two points moving towards the boundary of the space along two continuous curves, such that the Teichmüller distance between them approaches infinity while their length spectrum distance approaches zero. Unfortunately, the length spectrum function does not define a metric on the (unreduced) Teichmüller space of a Riemann surface with boundary. In the second part of this work, we introduce a modified length spectrum function that does define a metric on this space. We show that if two points are close with respect to the Teichmüller metric, then they are also close in the modified length spectrum metric. We also show that the converse is not true. Finally, we prove that the (unreduced) Teichmüller space of a Riemann surface of finite topological type with non-empty boundary is not complete under the modified length spectrum metric.

### On the rank of 2-primary part of Selmer group of certain elliptic curves

Author:KWANG HYUN KIMYear of Dissertation:2012Program:MathematicsAdvisor:Victor KolyvaginAbstract:Kolyvagin proved very remarkable results on Mordell-Weil groups and Shafarevich-Tate groups of certain elliptic curves when a given Heegner point P

_{K}has infinite order in his series of papers. He also extended his result to odd prime l-primary part of Selmer group of higher rank with the assumption of existence of non-trivial Kolyvagin system. In this thesis, we will follow his Euler system method and verify that his method also works to prove the result on the rank of 2-primary part of Selmer group of higher rank with Strong non-zero conjecture.### The Admissible Dual of SL(2) of the Dyadic Numbers

Author:Terence Kivran-SwaineYear of Dissertation:2011Program:MathematicsAdvisor:Carlos MorenoAbstract:The admissible dual of SL

_{2}(Q _{2}) is constructed uniformly, based on a method adapted from the the theory of cuspidal types of GL_{2}(F).