**Mathematics Senior Thesis Presentations**

**Spring 2011**

**Tuesday, May 3 ****(****Warner 202)****:**

* 3:00 p.m. Kim Ammons*

*Brooks’ Theorem of Graph Coloring and Some of Its Extensions *

We introduce the idea of vertex coloring in graph theory and present Brooks’ Theorem, a famous theorem which relates the chromatic number of a graph to its maximum degree. We will discuss why the theorem is so widely studied and why it lends itself to so many extensions. We will present a number of these extensions, including relations to list coloring and triangle-free graphs, and touch upon other work that has been done on the theorem.

* 3:30 p.m. Donny Dickson*

S*perner’s Theorem, Intersecting Set Pair Systems and Applications in Extremal Graph Theory*

An antichain is a system of subsets in which no subset is contained in any other. Sperner’s Theorem provides a definitive restriction on how many subsets can exist in an antichain. The concept of an antichain, however, can be generalized into more complex relationships between subsets. In this talk several of these systems, including Intersecting-Set-Pair systems, are presented with corresponding theorems on their size restrictions. Along with the proofs of these theorems, we will use many of these inequalities to yield interesting results relating to hypergraphs.

* 4:00 p.m. Kent Diep*

*Mathematical Modeling of Hepatitis B*

* *

* *Globally, there are an estimated 350 million individuals who are infected by the Hepatitis B virus. Thus, it is important to understand its transmission behavior in the future. Many mathematical tools exist for analyzing transmission dynamics of diseases. In this thesis presentation, we will first introduce a number of frequently used compartmental models. Here, a closer look at different parameters and assumptions will help us understand what is needed to design such disease models, and the kind of analyses that are necessary to understand their long-term behavior. Different practical examples will help illustrate these concepts. We will look at a recent Hepatitis B model for China that was developed by Zou, Zhang and Ruan in 2009. Based on the earlier concepts, we will see how they are applied to this specific model.

**Wednesday, May 4 ****(Warner 203)****:**

* 4:15 p.m. William Noble*

*Peano’s Axioms and Mathematical Nominalism: Recapturing Truth from Math’s Existential Crisis.*

Peano’s Axioms are five simple statements that can be used to prove propositions about the natural numbers that are usually themselves taken as axiomatic. In this talk, we will present Peano’s Axioms and one problem in the philosophy of math that they might help to solve.

* 4:45 p.m. Matt Sunderland*

*Perturbed Markov Processes and the Evolution of Social Institutions*

Classical game theory solution concepts such as the asymmetric Nash bargaining solution require extravagant assumptions about agents’ level of rationality and degree of common information. Can we reproduce these higher-rationality results in low-rationality environments? This talk combines topics from graph theory and probability theory.

* 5:15 p.m. Rob Nicol*

*The Geometry of Musical Chords*

The disciplines of math and music have been linked throughout human history, but only in the past decade has a viable model been developed that gives mathematical structure to musical chords and relationships between them. This model, known as “The Geometry of Musical Chords,” was introduced by Princeton music professor Dmitri Tymoczko in 2006 as a means to analyze “musical distance” between chords, effectively giving composers a map of possible directions to pursue in a composition. In the model, chords are represented by points in generalized Euclidean spaces, called orbifolds. We will first build the machinery necessary to define an orbifold, then apply this structure to musical chords following Tymoczko’s model.

**Thursday, May 5 ****(****Warner 202)****:**

* 3:00 p.m. Judy Jiao*

*Optimal Design of Queueing Systems*

Queues are an inevitable part of our lives. While waiting in a long line to pay for our groceries usually bring nothing much more than private annoyance, the cost of waiting for machines to be repaired, or waiting the assembly line to finish one process before the next commences can be considerable. Queueing theory is the study of waiting in these systems by making ppropriate assumptions about how customers arrive and how the facility operates. Queueing design also integrates economic concepts such as marginal cost, marginal utility, and social v.s. individual optimization. This talk will be accessible to those who are comfortable with basic calculus and statistics.

* 3:30 p.m. Armaan Sarkar*

*Intelligent Search Algorithms and the Singular Value Decomposition*

The aim of this talk is to discuss `smarter’ search algorithms. We humans are very good at making connections between different pieces of information. A large part of this is because we tend to remember these pieces in terms of the concepts they address rather than the specific of the words used to describe them. Latent Semantic Indexing is a computer search algorithm that attempts to mimic this ability. We will discuss the math used to achieve this, in particular the Singular Value Decomposition, and how it applies to the algorithm.

* 4:00 p.m. Nick Tkach*

*Hilbert’s Tenth Problem*

Hilbert’s Tenth Problem asks for an algorithm to determine whether a given Diophantine equation has solutions in integers; in this talk, we will give a condensed version of a proof that no such algorithm exists. Along the way, we will encounter some interesting results regarding Diophantine sets and recursive functions.