Thesis Exchange

From MathClub

Theses in the math department come in a wide variety. Some seniors write expository papers on specialized fields, some write original mathematical research papers, and others extend and elaborate on results in the mathematical literature.

The math majors (Class of 2007) are a tight-knit group, and there is a lot of interest in each other's work. This page is meant to allow graduating seniors to share their independent work with one another and with the community at large.

We encourage every senior who is interested to send the title, a brief description (3-4 sentences), and a pdf file of their thesis to either dhl@ or mathclub@ and it will be added to the list.

Class of 2007 senior theses

• Broderick, Tamara : Construction of a pairwise Ising distribution over a large state space with sparse data

This thesis looks at methods for improving feature-based maximum entropy. In the most relevant application, it brings the running time for modeling forty retinal neurons down from about 2 weeks to less than 2 minutes. For a more thorough description, also see Link

• Caraiani, Anna

• Chowdhury, Atoshi : The Brauer group of a local field and Brauer-Severi varieties: an explicit approach, File

We give an explicit, noncohomological computation of the Brauer group of a local field, then discuss Brauer-Severi varieties and sketch a proof that they satisfy the Hasse principle.

• Creighton, Robert

• Halpern-Leistner, Daniel : On the Algebraic Structure of Dynamical Systems, File

My thesis constructs algebraic and geometric representations of abstract dynamical systems, systems consisting of a Boolean algebra and an endomorphism. Some applications to the theory of generalized measures of information and to the theory of algebraic invariants for dynamical systems are discussed.

• Izmailov, Ilia

• Kamburov, Nikola : On the Distribution of Points with Coprime Integer Coordinates. Applications to a Problem Involving Continued Fractions, File

In my thesis I look at the limiting distribution of continued fraction digits of the coordinates of a discrete subset of the unit square. The subset in question corresponds to the set of points in the plane having coprime integer coordinates. The problem was motivated while studying a certain ergodic-theoretical aspect of the canonical extension of the Gauss transformation.

• Kaplan, Nathan : P-adic Integration and Subrings of Zn, File

In this paper we will attempt to count the number of subrings of Z^n of index k by computing certain p-adic integrals. We will begin by presenting background material on zeta functions of groups, Tauberian theorems, p-adic numbers and p-adic integration, and blow-ups. We will also present work of Igusa and Denef related to our basic question. At the end of this paper we answer our question for n=3 and present partial results for n=4 and n=5.

• Khandker, Zuhair : Real-space perturbative renormalization-group procedure on fractal lattices: Random antiferromagnetic Heisenberg model, File

The Heisenberg Spin-1/2 Hamiltonian with strong random couplings has been closely studied in one dimension, and some extensions to higher integer dimensions have also been made. It appears that a jump in dimension from a one-dimensional lattice to a two- or three-dimensional lattice is accompanied by changes in the system's fixed-point behavior in the limit of vanishing energy. In order to explore the effect of dimensionality on a system's low-energy behavior, we study a family of lattices, which we have called the ST Family, consisting of the log(3)/log(2) -dimensional Sierpinski Triangle and three one-dimensional lattices that geometrically resemble the Sierpinski Triangle. We use a real-space perturbative renormalization-group procedure originally introduced by Dasgupta and Ma to study the low-energy flow properties of the lattices in the ST Family. We find that dimensionality is in fact the primary criterion for distinguishing low-energy behavior in the ST Family: Although the lattices in the ST Family resemble one another in shape, the one-dimensional lattices share important low-energy characteristics, like an infinite-disorder fixed point, that the higher-dimensional Sierpinski Triangle does not share.

• Lin, David

• Nuer, Howard

• Petrow, Ian : Numerical Evidence for a Conjecture of Hooley, File

My thesis concerns class numbers of binary quadratic forms. In the case of negative discriminant forms, much is known and the theory is well-established. However, many questions about positive discriminant quadratic forms are very difficult. My paper investigates a conjecture made in 1984 on asymptotics of the average size of the class number for the positive case. It includes the first ever numerical verification of such a conjecture.

• Pierrehumbert, Anna

• Shen, Jian

• Silberstein, Aaron : Moduli Problems for non-Hermitian Symmetric Spaces, File

We give a brief introduction to descent theory and a general procedure for constructing sets of polarized abelian varieties with given maps into their endomorphism algebras through group quotients; these are first steps toward understanding Galois representations associated to automorphic forms on GL₂(K) where K is an imaginary quadratic number field.

• Speh, Peter

• Werness, Brent : Structural Theorems on Tournaments, File

My thesis presents a few original results in the structural theory of a class of directed graphs known as tournaments. In it I prove several total and partial characterizations of various excluded subgraph questions.

• Zipkin, Joseph

Math Related Theses from other departments

• Li, Tianhui (Michael) - Computer Science : Automated Reduction of Nonstandard to Standard Analysis, File

Nonstandard Analysis is a twentieth century formalization of the notion of infinitesimals, which are not rigorous in the usual formulations of Analysis. We use a system of Nonstandard Analysis called IST, a logical extension of ZFC that admits the predicate `standard' and three extra axioms, Internalization, Standardization, and Transfer. A reduction algorithm is given that translates statements and proofs in IST into ones of ZFC which are equivalent to the original statements in IST. Unfortunately, my thesis lacks much motivation: it tackles the problem from a purely logical perspective. The reader interested in the analytic motivation should read Edward Nelson's (unfinished as of 2007) book on nonstandard analysis

• Kwon, Yeong Dae - Physics : A Study of Confidence Intervals for the Daya Bay Project, File

A formal theoretical justification is given for the χ² method of determining the confidence interval for the Daya Bay neutrino-oscillation experiment. A new analytic method of calculating the interval is devolped and used to compute the dependence of the interval on various systematic uncertainties.