**Mathematical Modeling in Ecology: Simulating the Reintroduction of the Extinct Passenger Pigeon (Ectopistes migratorius)**

Professor Alex Capaldi

The Passenger Pigeon (Ectopistes migratorius) was an iconic species of bird in eastern North America that comprised 25-40% of North American avifauna. Passenger Pigeons went extinct in 1914 due to excessive hunting over the previous 50 years. Current research aims to de-extinct the Passenger Pigeon and someday release the species into its historic range. To determine under which conditions a Passenger Pigeon could survive a reintroduction into a natural habitat, we used two types of models. First, we used a system of delay differential equations to explore the relationship between the young pigeon, adult pigeon, nest predator, and raptor populations. The model incorporates logistic population growth, an Allee effect, and a Holling Type III functional response. Next, we developed a spatially explicit, agent-based model to simulate the population dynamics of the Passenger Pigeon in a number of present-day forest environments. The model incorporates the following stochastic processes: varying availability of food sources, reproduction, and natural death of the Passenger Pigeon. Bio-energetics, tree distributions, and other ecological values were obtained from literature. Results from our simulations suggest that the Passenger Pigeon could survive a reintroduction into a natural environment.

**Noise-Induced Stabilization of Stochastic Differential Equations**

Professor Tiffany Kolba

The phenomenon of noise-induced stabilization occurs when an unstable deterministic system of ordinary differential equations is stabilized by the addition of randomness into the system. Noise-induced stabilization is quite an intriguing and surprising phenomenon as one’s first intuition is often that noise will only serve to further destabilize the system. In this paper, we investigate under what conditions one-dimensional, autonomous stochastic differential equations are stable, where we take the notion of stability to be that of global stochastic boundedness. Specifically, we find the minimum amount of noise necessary for noise-induced stabilization to occur when the drift and noise coefficients are power, exponential, or logarithmic functions.

**Pattern Avoidance in Reverse Double Lists**

Professor Lara Pudwell

Pattern avoidance is a branch of combinatorics that arose in 1968 when Donald Knuth began studying stack sorting. One central problem in pattern avoidance is finding the number of permutations of length n that avoid a specific pattern . We expanded this problem to reverse double lists, or lists built by combining a permutation with its reverse. We computed the number of reverse double lists of each length that avoid patterns of up to length four and then conjectured and proved formulas to explain these sequences.

**Pattern Avoidance in Double Lists**

Professor Lara Pudwell

The motivation for this paper is to continue combinatorics research in the area of pattern avoidance. We have constructed a subset of words to study called “double lists” which are based on the standard permutations used in counting pattern avoiding lists. We will begin this paper with an introduction to pattern avoidance for those unfamiliar. However, the majority of this paper will focus on our own research of pattern avoidance within double lists.

**The Influence of Risk-Taking Behavior on The Evolution of Infectious Diseases**

Professor Daniel Maxin

Individuals facing an infectious disease, consciously or not, evaluate the contamination risk and engage in or avoid risky situations or behaviors. The risk-taking behavior of any individual is probably a very complex functional response that depends on many factors such as: morbidity of the disease, infectiousness, mode of transmission, existence and eﬃcacy of treatment, etc. There are multiple studies that show that individuals have dynamic responses to transmission risks that change with varying circumstances. For example, the advent of antibiotics makes all of us less concerned with exposure risks to common pathogens. Nobody is worried about getting tick bites in the woods, since a course of penicillin would cure a possible exposure to Lyme disease. To the contrary, knowledge about the severity of disease (measured by number of infectious people, virrulence, lack of eﬀective treatment) may cause susceptible individuals to be more cautious and reduce their exposure risk. From these remarks we can talk about two general ways in which behavioral attitudes inﬂuence the spread of a disease:

**Estimating Option Prices with Heston’s Stochastic Volatility Model**

Professor Hui Gong

Options are a type of financial derivative. This means that their price is not based directly on an asset’s price. Instead, the value of an option is based on the likelihood of change in an underlying asset’s price. More specifically, an option is a contract between a buyer and a seller. This contract gives the holder the right but not the obligation to buy or sell an underlying asset for a specific price (strike price) within a specific amount of time. The date at which the option expires is called the date of expiration.

**Mathematical Modeling in Ecology – What Killed the Mammoth?**

Professor Alex Capaldi

During the Paleolithic Period, mammoths, as well as other mega mammals, went extinct. The exact reasons for this have been debated for some time, and current hypotheses point to causes such as disease, climate change, over hunting by humans, or some combination thereof. However, recent developments in mathematical ecology may be able to elucidate the matter. A recent study has given strong evidence in support of the hypothesis that the Neanderthals’ extinction was due to strong competition from modern humans. The goal of this project is to follow in the footsteps of the Neanderthal study and consider the similar question as to whether the mammoth’s extinction was anthropogenic.

Prerequisites: At least one of the following: differential equations, statistics, programming experience.

**Estimating the Volatility in the Black-Scholes Formula**

Professor Hui Gong

Black-Scholes formula has been a huge success since its introduction. However, the formula has a component of the volatility, which is not deterministic and available immediately for application of this formula. We can estimate this volatility via multiple models or methods, such as parametric approaches with different probabilistic models, nonparametric approaches and time series models approaches, such as smoothing techniques. We will derive these estimates and incorporate into the Black-Scholes formula.

Prerequisites: an introduction of statistics, understanding the basic probability distributions (uniform, normal, Chi-square, etc.), the concepts of mean, variance, and correlation.

**Graph Labelings**

Professor Zsuzsanna Szaniszlo

**Paper: 4−Equitable Tree Labelings**

In this project we will look at edge-vertex graphs and investigate a certain assignment of numbers to the edges and vertices. These so-called labeling problems were introduced in the 1960s as possible tools for solving graph decomposition problems. Since then different labelings were studied for their own sake and for other applications as well. In the summer project, we will investigate when we can distribute labels (numbers) equally in a graph. Answering this question for trees would settle the famous graceful tree conjecture.

Prerequisite: a proof-based course

**Quandles and Generalized Colorings of Knots**

Professor Paul Drube

**Paper: A Partial Ordering Of Knots**

Knot theory is a branch of topology that studies how one may embed a circle within three-dimensional space. Since knots may become quite complicated, much of modern knot theory involves the development of invariants that can distinguish between different equivalence classes of knots. One basic class of knot invariants is the Fox n-colorings of knots, which count the ways that one may appropriately “decorate” the strands of a knot diagram with elements of a cyclic abelian group. In recent years, the mathematical structure underlying these Fox n-colorings has been greatly generalized utilizing abstract binary operations known as quandles. After a brief introduction to knot theory, this project will explore basic properties about quandles and their associated knot invariants, hopefully developing interesting results about the generalized colorability invariants of specific classes of knots.

Prerequisites: linear algebra; an introductory proof course or basic abstract algebra (group theory) may also be helpful.

**Vertical transmission in two-sex epidemic models with isolation from reproduction**

Professor Daniel Maxin

**Paper: Is More Better? Higher Sterilization of Infected Hosts Need Not Result in Reduced Pest Population Size**

Vertical transmission is the transmission of an infection from mother to child at birth. During the academic year 2008-2009, Tim Olson and Adam Shull (students at Valparaiso University) worked on a project with me to study the influence of isolation from reproduction on sexually transmitted infections with vertical transmission. One of the questions addressed in their project was the behavior of the disease in the case of 100% vertical transmission (i.e. all newborns from infected mothers acquire the disease). A surprising result showed that, under several conditions on the vital parameters, the isolation from reproduction may prevent a Susceptible Extinction situation (a case when the disease eliminates the entire healthy population) even when the infection rate is large. For simplicity the model did not separate the individuals by gender (to keep the number of equations low). Our project consists of establishing a two-sex version of this model that follows the dynamics of females and males. The main objective is to investigate if the result described above holds in the more realistic framework of gender structured models and whether the result is independent from various types of mating functions commonly used in the literature. A secondary objective is to improve the original model with features that were not included so far such as diseases with recovery or temporary isolation from reproduction.

Prerequisites: Differential Equations

**Generalized Pattern Avoidance in Trees**

Professor Lara Pudwell

**Paper: Non-Consecutive Pattern Avoidance in Binary Trees**

A current hot topic in combinatorics is enumerating structures that avoid certain patterns. In the 2010 VERUM program, Gabriel, Peske, and Tay studied contiguous pattern avoidance in ternary trees. In the 2011 VERUM program, Dairyko, Tyner, and Wynn studied non-contiguous pattern avoidance in binary trees. There are a number of variations on tree-pattern avoidance that remain to be explored by modifying what types of trees or what types of patterns are considered. We will consider a new type of tree pattern that is a hybrid of contiguous patterns and non-contiguous patterns. In particular, counting problems in the 2010 project always resulted in algebraic generating functions, while counting problems in the 2011 project always resulted in rational generating functions. We will try to characterize whether hybrid patterns fall under the algebraic or rational paradigm and extend our work from there.

Prerequisites: linear algebra, or another proof-based course; a course in combinatorics, discrete math, or elementary graph theory would be helpful.

**Mathematical Models of Infectious Disease**

Professor Alex Capaldi

**Paper: Mathematical Models of Infectious Diseases**

In a world where infectious diseases, such as the H1N1 flu, bird flu or SARS, pose a tangible public health risk, it is vital to understand the dynamics of outbreaks. We will see how to gain information about epidemics by using an array of mathematical tools including differential equation models, stochastic models, and network models. Different control strategies for containing diseases such as vaccination and quarantine and how they can be optimized using mathematics will be introduced. An infectious disease topic from current events (such as the swine flu pandemic, the whooping cough outbreaks in California, or even the mythology of the spread of vampires in a human population) will be chosen to be studied in depth.

Prerequisites: At least one of the following: differential equations, statistics, programming experience.

**Pattern Avoidance in Trees**

Professor Lara Pudwell

**Paper: Non-Consecutive Pattern Avoidance in Binary Trees**

A current hot topic in combinatorics is enumerating structures that avoid certain patterns. In 2008, Rowland defined contiguous pattern avoidance in binary trees. In the 2010 VERUM program, Gabriel, Peske, and Tay extended Rowland’s results to ternary trees. For all this previous work, the trees were rooted and ordered, and the tree patterns were contiguous. Any of these three conditions can be relaxed to produce a related, but new and as-yet-unstudied counting problem. We will begin by considering trees that contain non-contiguous tree patterns and work to generalize our results from there. Prerequisites: a course in combinatorics, discrete math, or elementary graph theory would be helpful.

**Computer Monitored Problem Solving Dialogues**

Professor Michael Glass

**Paper: Computer Monitored Problem Solving Dialogues**

Is it possible to monitor small groups of students solving a problem online? A common instructional method in mathematics classrooms is to have students work together exploring some phenomenon while the teacher walks around the room monitoring progress and assisting as needed. As instruction moves online, two issues are how to adequately replace the manipulatives-and-worksheet physical problem-solving environment and how to get the computer to “see” what the teacher sees by looking over shoulders, using clickstream data and the students’ conversation. Research questions include sensing student affective states (e.g. bored, engaged, confused), student cognitive states (e.g. what aspects of the math do they understand), and degree and quality of collaboration.

Prerequisites: strong programming abilities in a high level language.

**The logistic two-sex model without pair-formation**

Professor Daniel Maxin

**Paper: The Impact of Sexually Abstaining Groups on Persistence of Sexually Transmitted Infections in Populations with Ephemeral Pair Bonds**

It has been shown that the isolation from reproduction may induce a locally asymptotically stable disease-free equilibrium in an endemic situation. For human populations, a gender-structured model includes single individuals (females and males) and couples. In animal populations however, various species do not form stable pairs, and the mating is a consequence of direct encounter among individuals of different genders. The main objective of this project is to develop and analyze a two-sex logistic model without pairs and to extend this model to an epidemic model and verify if and under what conditions previous results are still valid. A second objective is to analyze in more detail the mathematical and biological properties of various fertility functions both theoretical and against real data.

Prequisites: Differential Equations

**Generalized primes on the Mosaic of an Integer**

Professor Rick Gillman

**Paper: Mosaic Arithmetic**

The mosaic of an integer is the configuration of primes obtained by repeated applications of the Fundamental Theorem of Arithmetic to a positive integer *n* and any composite exponents. The concept of the mosaic was introduced by Mullin, in a series of papers in the early 1960s. Various arithmetic functions dependent only on the primes in the mosaic were investigated by Girse and Gillman. During the summer of 2007, a team of undergraduate research students investigated this same structure and developed several potential concepts of divisors, allowing the set of arithmetic functions defined on mosaics to be expanded greatly. This team will continue to develop these notions of divisibility, eventually leading to more fully developed theory of mosaics.

Prerequisites: an elementary number theory course would be helpful.

**Pattern Avoidance in Ternary Trees**

Professor Lara Pudwell

**Paper: Pattern Avoidance in Ternary Trees**

For this project we will consider rooted ordered trees avoiding other trees. In 2008, Rowland defined pattern avoidance in binary trees (i.e. trees where each vertex has either 0 or 2 children). The standard combinatorial sequences appear when enumerating such trees, and there exist bijections between these trees and other common combinatorial objects. There is also a method to determine equivalence classes of binary trees based on the number of trees that avoid them. This team will extend these known results about pattern avoidance to ternary trees (i.e. trees in which each vertex has 0 or 3 children).

Prerequisites: linear algebra, or another proof based course; a course in combinatorics, discrete math, or elementary graph theory would be helpful.

**Groundwater Flow**

Director: Ken Luther

**Paper: Self-Influencing Interpolation in Groundwater Flow**

In this project we will investigate analytic (closed form) solutions to three dimensional steady state groundwater flow problems. While numerical solutions to groundwater flow problems are more commonly used in practice, especially in the three-dimensional realm where analytic solutions are cumbersome or even nonexistent, the construction of some analytic solutions is essential so that numerical models can be validated. The specific problem at hand will involve flow to a well in a stratified aquifer, and/or interaction between a horizontal well and a surface water body.

Prerequisites: multivariable and vector calculus, linear algebra, and differential equations. Exposure to partial differential equations is a plus, and experience with MATLAB is a double plus.

**Functions on the Mosaic of n**

Director: Rick Gillman

**Paper: Mosaics: A Prime-al Art**

Any integer n can be uniquely factored into a product of prime powers. Each of the resulting exponents greater than 1 can, in turn, be factored into a product of prime powers. Iterating this process until there are no composite exponents results in an array of numbers called the mosaic of n. In this project we will define new arithmetic functions on the mosaic of n and investigate their arithmetic and algebraic properties.

Prerequisites: linear algebra or another proof-oriented course. A course in elementary number theory would be helpful.

**Understanding Mathematics Tutoring Dialogue (Computer Science project)**

Director: Michael Glass

**Paper: Computerized Tutoring**

As part of building dialogue-based computer tutors for mathematics, this project works on developing the techniques for computer understanding of student utterances. During the course of a tutoring conversation students can perform discourse actions such as checking whether some idea is true, asking a question, expressing confusion, and so on. Examining transcripts of tutoring sessions, this project will work on software methods for guessing the intentions behind a student’s utterance. This summer’s experiments may involve both numerical and symbolic methods, such as statistical models, latent semantic analysis, and finite-state machines.

Prerequisites: one mathematics class above calculus. Students should have strong programming abilities including experience in a high level language, for example Python or List.

**Distributions of Interest Disfor Quantifying Reasonable Doubt and their Applications**

Director: Professor James Caristi

**Paper: Distributions of Interest for Quantifying Reasonable Doubt and Their Applications**

The concept of reasonable doubt is a standard of our legal system; however, it is a standard that is not well defined. Differences in the way reasonable doubt is applied in different courts and states, as well as ambiguities in its different definitions, suggest that the standard puts pressure on due process and equal protection concerns. This paper explores probability distributions that will aid in the understanding of the American legal system as it is today, what reasonable doubt means under this system, and how reasonable doubt should be defined.

**Systems of Matrix Equations**

Director: Professor Patrick Sullivan

**Paper: On the Properties A _{m,n} for Subspaces of C^{kk}**

his project will study properties of complex matrices. The students will explore what subsets of matrices have certain properties related to rank one matrices. We will be studying when arrays of matrices can be solved simultaneously as rank one matrices up to a certain equivalence.

**Graph Labelings**

Director: Professor Zsuzsanna Szaniszlo

**Paper: L(3, 2, 1)-Labeling of Simple Graphs**

An L(3,2,1)-labeling is a simplified model for the channel assignment problem. It is a natural generalization of the widely studied L(2,1)-labeling. An L(3,2,1)-labeling of a graph G is a function f from the vertex set of the graph to the set of positive integers such that for any two vertices x,y, if d(x,y)=1 then | f(x)-f(y) | ≥3; if d(x,y)=2, then | f(x)-f(y) | ≥2; and if d(x,y)=3, then | f(x)-f(y) | ≥1. The L(3,2,1)-labeling number k(G) of G is the smallest positive integer k such that G has an L(3,2,1)-labeling number for paths, cycles, caterpillars, n-ary trees, complete graphs and complete bipartite graphs. We also present an upper bound for k(G) in terms of the maximum degree of G.

**Mathematical Models in Traffic Assignment and Congestion Pricing**

Director: Professor Lihui Bai

**Paper: A Genetic Algorithm For The Minimum Tollbooth Problem **

This project uses rigorous mathematical models to study traffic assignment in urban transportation networks. Traffic assignment distributes vehicles in a transportation network so that certain criterion are satisfied. The study of traffic assignment models can be used in traffic congestion management, where traffic planners want to minimize the total travel delay for a given transportation network.

**Crystallographic Groups**

Director: Professor Kimberly Pearson

**Paper: Virtually Cyclic Subgroups of Three-Dimensional Crystallographic Groups**

An enumeration of the virtually cyclic subgroups of the three-dimensional crystallographic groups is given. Additionally, we offer explanations of the underlying group theory and develop several exclusion theorems which simplify our calculations.

**Power Distributions in Weighted Voting Systems**

Director: Professor Rick Gillman

**Paper: Using Sets of Winning Coalitions to Generate Feasible Banzhaf Power Distributions**

Given a weighted voting system, the Banzhaf Power Index can be used to determine the power distribution of the individual voters. We are interested in the converse of this problem: given a collection of voters, can a weighted voting system be constructed which has a prescribed power distribution? This problem has been solved for a system with four voters, but still open in more general settings.