**Rachael Fountain (advisor John Judge):**

#### Exploring the Variability in Music Melody Sequences

Abstract: We all know that music is so much more than just the notes, pitches, and beats we hear; in fact there are surprising connections between music and mathematics. In this presentation we will explore the relationship between musical melodies and statistics by looking into the concepts of variability and correlation. I will then apply concepts to analyze several pieces of music. I will also discuss “The Beatles Rule” and how it is established from the idea of variability in music melody sequences.

## Jacob Goudreau (advisor Brian Jennings)

#### The Mathematics of Quantum Entanglement

Abstract: In this talk I will discuss how the concept of quantum entanglement, from a mathematical point of view, follows naturally from the mathematics of quantum theory. In particular we will look at the example of an entangled two-state system. Any student who is familiar with the basic concepts of linear algebra will be able to follow along.

## Zachary Lancto (advisor Karin Vorwerk)

#### Fibonnaci Matrices

Abstract: The 2x2 Fibonacci matrix is well-known to mathematicians. I will be discussing the properties of this matrix and describe the underlying graph. I will also present some results and insights I gained when extending the concept of Fibonacci matrices to higher dimensions.

## Jessica Young (advisor Maureen Bardwell)

#### Cayley's Theorem And Beyond

Abstract: Cayley's Theorem is a very well known theorem in abstract algebra which states that every group (G, *) is isomorphic to a subgroup of the group of permutations of the set G. Usually this is applied to finite groups, however this theorem is also very useful in looking at infinite groups. I will be exploring further applications of Cayley's Theorem and showing various examples of how Cayley's Theorem can help to solve Birkhoff's problem of classifying all ordered sets whose group of permutations is a lattice ordered permutation group.