## Mathematics Department

Undergraduate Research Symposium

Spring 2016

## May 3, 12:30-2:45pm - Room LCB 323

12:30-12:45 Tyler McDaniel

Mentor: Braxton Osting

Utah’s Pathways to Higher Education: A Quantitative, Critical Analysis

12:45-1:00 Chong Wang

Mentor: Don Tucker

Eradicating Ebola

1:00-1:15 Dan Armstrong

Mentor: Sean Lawley

Methods for Modeling Neurite Growth Driven by Vesicular Dynamics

1:15-1:30 Curtis Houston

Mentor: Jyothsna Sainath

Mapping Counts of Death in League of Legends

1:30-1:45 Yushan Gu

Mentor: Firas Rassoul-Agha

Rare Events for Non-Homogenous Markov Chains

1:45-2:00 Naveen Rathi and Gerardo Rodriguez-Orellana

Mentors: Owen Lewis and Leif Zinn-Björkman

A Mechanical Model for Simulating the Cell Motility of a Visoelastic Cell

2:00-2:15 Nathan Willis and Olivia Dennis

Mentors: Owen Lewis and Leif Zinn-Björkman

Mathematical Model of Cell Motility

## Abstracts:

**Tyler McDaniel**

Mentor: Braxton Osting

Utah’s Pathways to Higher Education: A Quantitative, Critical Analysis

**Abstract:**Using data from the 2008 cohort of Utah high school graduates, this project analyzes the effect of demographic factors (race, class, mobility, language, geography) and past achievement (ACT scores, AP scores, GPA) on the individual’s likelihood of attaining success in higher education (college entry, GPA, graduation). Data is obtained from the Utah System of Higher Education, and consists of 41,303 observations (students). The goal is to describe college pipelines in terms of access for racial, class, mobile, language-learner and geographic groups. What groups are advantaged in the college-application process? Equity will be assessed using a critical pedagogy framework, in which education has the potential to liberate students from systemic oppression (Friere, 1970, Harris, 2014).

Principal Components Analysis (PCA) was used in order to choose variables of importance and model underlying factors. Principal Components Analysis (PCA) is appropriate for reducing a dataset in order to analyze its most important variables. ACT Scores were the most important variable in all PCA models. Further, ACT Math, Science and Composite scores were consistently the variables with the highest loadings in our models. This result affirms education literature stating that math and science scores are particularly important in the college-going process.

Additionally, college pathways will be described using a Random Forest (RF) decision tree algorithm. The RF model has been used widely across academic fields and returns the predictive power of variables included. Further, the RF model is predictive, using random samples of predictor variables to generate decision trees that form “forests.” The RF model is ideal for college access data because it has robust algorithms for dealing with missing data (Hardman et. al, 2013).

**Chong Wang**

Mentor: Don Tucker

Eradicating Ebola

**Abstract:**Build a realistic, sensible, and useful model that considers not only the spread of the disease, the quantity of the medicine needed, possible feasible delivery systems (sending the medicine to where it is needed), (geographical) locations of delivery, speed of manufacturing of the vaccine or drug, but also any other critical factors I consider necessary as part of the model to optimize the eradication of Ebola.

**Dan Armstrong**

Mentor: Sean Lawley

Methods for Modeling Neurite Growth Driven by Vesicular Dynamics

**Abstract:**Neurite growth requires both membrane expansion via vesicle exocytosis and cytoskeletal dynamics. Mathematicians have modeled neurite growth by simulating the dynamics of vesicle motion and microtubule interaction at the boundary of the growing neurite. We examine different methods, both stochastic and deterministic, for modeling neurite growth and compare the results. Stochastic models use a coarse-graining method to eliminate intermittent dynamics and derive a single SDE that describes vesicle motion. In past studies, this coarse-grained SDE is derived using vesicle motion in two-dimensional cells. We use a method that produces a more accurate SDE because it is derived using vesicle motion in a three-dimensional cell. Using this SDE we generate models of growing neurites, which predict different neurite growth regimes depending on cytoskeletal dynamics.

**Curtis Houston**

Mentor: Jyothsna Sainath

Mapping Counts of Death in League of Legends

**Abstract:**In the video game League of Legends, player deaths greatly impact the tide of each match. In this project we take match data from League games and use a bootstrap method to find counts of kills across the map in order to better understand where players are likely to die. We then separate kills by time in order to consider how these counts change throughout the course of the game and across the map. In particular, we find that kills tend to move toward the middle lane of the map and spread further apart up and down the lanes as the game progress.

**Yushan Gu**

Mentor: Firas Rassoul-Agha

Rare Events for Non-Homogenous Markov Chains

**Abstract:**This project is about simulating rare events for non-homogenous Markov Chains. When a fair coin is flipped, the probability that we get heads is 1/2. When we flip this fair coin n times, the expected number of heads is n/2. Getting 0.7n heads is a rare event for this model. There is a very small probability that this rare event occurs. Therefore, a huge number of simulations is required in order to generate such a rare event. Using the theory of large deviations and the notion of entropy one can calculate the distribution of the sequence of iid coin tosses, conditioned on the rare event. Then one can use this to generate rare events. Now, every sample is rare event. The same can be done for homogenous Markov Chains. However, in applications such as climate models, one often has non-homogenous Markov Chains. The final goal of this project is finding the distribution of the process conditioned on rare events, for non-homogenous Markov Chains. More precisely, we consider two Markov Chains with state space {0, 1} and the Markov Chain that alternates between the two. We will find the distribution of this alternating process, conditional on it having an unusual number of ones.

**Naveen Rathi and Gerardo Rodriguez-Orellana**

Mentors: Owen Lewis and Leif Zinn-Björkman

A Mechanical Model for Simulating the Cell Motility of a Visoelastic Cell

**Abstract:**In modern biology, cell locomotion is a key issue that is under constant investigation by both theoreticians and experimentalists. By understanding how a cell’s internal rheology and active contraction, as well as interactions with its surrounding environment affect the speed of the movement, it is possible to gain deeper insight into the mechanics of cell locomotion. In this research, we developed a mechanical model of a viscoelastic cell and numerically simulate to determine how variations in the time-dependent interactions affect migration velocity. In particular, we investigated how the relative phase of time-dependent adhesion and contraction affects the speed of the modeled cell.

**Nathan Willis and Olivia Dennis**

Mentors: Owen Lewis and Leif Zinn-Björkman

Mathematical Model of Cell Motility

**Abstract:**Cell motility is a vital process in a wide array of biological contexts including immune response, embryonic development, and wound healing, as well as the spread of cancer cells. Following previous studies, we develop a one-dimensional partial differential equation which models a motile amoeboid cell by balancing internal body forces with drag against the underlying substrate. We numerically simulate this model using Finite Differences and the Forward Euler method. We investigate the profile and coordination of adhesion between the cell and substrate. Specifically, we are interested in how the coordination of adhesion relative to active contraction within the cell affects the behavior of a motile cell.