Senior Capstones

Ryan Becker - Games Played on Graphs

We begin by introducing a game in which chips are placed  on the vertices of a graph  and   can  be  moved  in  a  proscribed way to  reach  a  winning  configuration. After exploring this game, we use our knowledge of how it is won to define what is called the Riemann-Roch space of the graph. We conclude by examining several "nice" graphs for which we can explicitly construct the Riemann-Roch space.

Jim Bennett - Predicting College Football Rankings

Anyone who follows college football will tell you that the Bowl Championship Series, or BCS, is interesting to say the least. We can do a minimal amount of research and see that the BCS has fallen under much scrutiny in past years.  The voting polls, however, have not really been looked at extensively.  Can we use a few parameters from each game and predict which teams will be ranked in the top 25 by the voting polls?  Can we also use the results from the first question to find a bias among voters?  Our journey to answer both questions starts here.

Dave Braunlich - Mathematics of Limit Hold’em

Some say that poker is game of luck.  In the last ten years the “poker boom” has drastically increased its popularity.  However, a certain group of professionals has been exploiting the game’s mathematical principles to reap tremendous profits.  This project uses Limit Texas Hold'em to explore what these principles are.  Using expected value and probability theory, we look at some common post-flop situations to find and develop an optimal strategy to maximize profits.

Nathan Blyler - Searching for Sub-populations in Flowers Using the EM Algorithm

Virtually everyone likes flowers, especially when they bloom.  However, just because within one species of flower plants look the same, does not mean each individual plant has the same probability of blooming.  In nature, plants transition between periods of dormancy and flowering for reasons that are not completely understood. Previous literature suggests a link between the dormancy and flowering states of numerous plants, but this paper focuses on the wild flower Silene spaldingii.  After evaluating the appropriateness of the Expectation Maximization Algorithm, we use it to test the hypothesis of the existence of multiple sub-populations within the wild flower Silene spaldingii distinguished by the dormancy or flowering characteristics of a plant.

Jennifer Donoghue - Mathematical Modeling of HIV Infection in the Human Body

Everyone knows that HIV/AIDS is a serious global epidemic that results in the death of millions of people each year. Within the past few decades, a whole host of research has been conducted in order to learn more about the disease and develop ways to fight against it. In this project I describe how HIV affects the body and discuss some of the important recent findings about the disease. I then use this knowledge to build a mathematical model to simulate the effects of the infection on the body’s immune system and talk about why this is useful. The project will be sure to provide an answer to the question, “how is math actually useful in the real world?”

Margaret Flora - Where High School Algebra & Geometry End & Algebraic Geometry Begins

Many students wonder how math from high school relates to the math they do as college students.  This paper first refreshes basic ideas from high school algebra and geometry, such as degrees of curves and the Fundamental Theorem of Algebra. That leads to the exploration of a concept from the field of algebraic geometry, Bézout’s Theorem, and the criteria needed for the application of Bézout’s Theorem, which is beyond the scope of high school algebra and geometry. Concepts of discussion include intersections of curves at infinity and intersection multiplicities of curves.

Edison Geng - On the Maximum Size of Unrestricted Sumsets

Additive Combinatorics is a relatively new field in Mathematics with many interesting problems.  I have chosen to research about the quite unexplored field of Maximum Size of Unrestricted Sumsets.  Given a m-subset A of a finite Abelian group G, we have defined the sumset of A corresponding to two sets, a subset of Z, denoted as λ, and a subset of N0, denoted as H.  I am now interested in finding the maximum size of the sumset of m-subset:  A given set λ and a set H and denote that maximum size as  µ λ (G, m, H).  In particular, I focus my attention on the case where G=Zn, m=3 and H=s.  I have come up with a general relationship between s and µ λ (G, m, H) and provide  a proof in my paper.

Sterling Hallfors - Maximum Size of Weak Sum-Free Sets in Abelian Groups

A set A is weak sum-free  in an  Abelian  group  Zn, if given two integers  k and  l, the set  of integers  produced from the sums of k distinct elements  in A is disjoint  from the set produced from  the  sums  of l distinct elements.  This  research  reviews  previous  work  with  sum-free  sets, shows  the  differences  of this  research,  and  highlights  some  of the  optimal sets  used  as well as some exciting  results.

Margaret Kelly - Host-Parasitoid Interactions

This project explores how populations grow and interact in discrete time. One specific type of species interaction that we look at is called a host-parasitoid interaction, where one species, the parasitoid, grows on and lives off the other species, the host. The Nicholson Bailey Model shows how these species interact, but in contradiction to most natural interactions, is unstable for all parameter values.  We examine a way to stabilize the model, as well as show how the model can be used for biological control.

Amanda Kreuter - Facial Recognition: Not Quite “As Seen on TV”

Anyone who has watched a modern crime show might wonder how the police can match security camera footage to a database of mug shots. There are a multitude of criteria and calculations that go into matching faces correctly. The crux of the computation is in creating a feature vector that is specific to each face, then comparing vectors for closest matches. This project details the conditions needed to compute a successful facial recognition as well as one of many methods of creating a feature vector: the Discrete Cosine Transform.

Katie Timmerman- A Talk on Graph Theory: Finding Maximal Independent Sets

Many graph theorists battle with the problem of finding large independent sets within a graph. In my research, I discuss strategies for tackling this problem. An independent set is a subset of a graph G=(V, E) with V vertices and E edges such that no two vertices share an edge. My paper explores how the “Timmerman Algorithm" can be used to find the maximum size of an independent set. Further, I list examples, present a theorem, and discuss its applications to scheduling problems. In my conclusion I bring everything together to explain how the algorithm works and why it is useful.

Eric Weil- The Hunt for October: A Mathematical Examination of the Major League Baseball Playoffs

Baseball is America’s pastime, but over time there have been some changes.  Major League Baseball has changed the playoff format and the effects are not fully understood.  In 2012, an extra Wild Card was added.  In 2013, the Astros switched leagues.  Now, how many wins does a team need to expect to make the playoffs?  Win the World Series?  This research explores how expanded playoffs and realignment affect parity in the majors.  Using Markov chains, we look at theoretical teams’ likelihood of success in the playoffs.  The hunt for October begins here.