## Mathematics Department

Undergraduate Research Symposium

Spring 2015

## Session 1/2 - Monday May 4 - 1pm-3:10pm - Room LCB 222

01:00-01:30pm,

**Anand Singh and Braden Schaer**,

*A Model for Restricted Diffusion of Evoked Dopamine*

*Mentors:*Sean Lawley and Heather Brooks

*Abstract:*A simple gap diffusion model is commonly used for short, unaided intercellular transport. One prominent example is dopamine diffusion in neural synapses, where intercellular space is often a crowded environment consisting of extracellular networks, biological waste, macromolecules, and other obstructions. The traditional model fails to represent many unique characteristics of restricted diffusion in complex cellular environments. We compare existing mathematical models of gap and restricted diffusion against data acquired by fast-scan voltammetry of evoked dopamine in the striatum of rat brains. Finally, we propose a more physically realistic model that accounts for the spatial structure of this system.

(Supported by Mathematical Biology RTG grant)

01:30-01:50pm,

**Oliver Richardson**,

*Precise Timing in a Neural Field Model*

*Mentors:*Sean Lawley and Heather Brooks

*Abstract:*Building upon the the work of Veliz-Cuba et al, this project addresses building a system of differential equations that lead to the precise encoding of neural firing event timings into the weighting between neurons. However, instead of dealing with a discrete case and a global inhibitory neuron, we have generalized this to the continuous neural field setting, and built inhibition directly into the weight itself.

(Supported by Mathematical Biology RTG grant)

01:50-02:10pm,

**Marie Tuft**,

*Quantitative Analysis of Virus Trafficking in a Biological Cell*

*Mentors:*Sean Lawley and Heather Brooks

*Abstract:*For a virus to successfully infect a host cell it must travel from the cell wall to the nucleus by hijacking that cell's existing transport system. This motion occurs as two iterated steps: passive diffusion through cell cytosol and active transport along microtubule networks. An existing model shows that this process can be approximated as a stochastic differential equation in the limit as the number of microtubules goes to infinity. We propose a different model which reduces the complex viral trajectory to a simpler finite state Markov process. Preliminary results show this approximation to be superior to the existing model across several modes of comparison.

(Supported by Mathematical Biology RTG grant)

02:10-02:30pm,

**Hitesh Tolani**,

*Dependence of Disease Transmission on Contact Network Topology*

*Mentors:*Braxton Osting and Damon Toth

*Abstract:*Despite much attention in recent years, there are a number of open questions related to epidemic dynamics for many infectious diseases, such as measles and ebola. Consequently, intervention strategies for these diseases are inadequate. As transmission occurs through direct contacts between infected and susceptible individuals, epidemic models must include properties of the host population's social structure. For example, individuals with more social contacts are more likely to transmit (or receive) an infection. It follows that the network topology of a social network has an impact on epidemic dynamics, such as growth rate and final outbreak size. In this work, we studied the influence of network topology on epidemic dynamics using agent-based simulations on synthetic social networks that were constructed using the contact history of students at a local elementary school. (No students were hurt in these simulations.) I'll discuss the results of these simulations and conclude with some preliminary results on improved intervention strategies.

02:30-02:50pm,

**Yuji Chen**,

*The Google rank page*

*Mentor:*Peter Alfeld

*Abstract:*I will describe and discuss the original algorithm that Google used to rank web pages.

02:50-03:10pm,

**Alex Beams**,

*Implications of Antibiotic Use for Co-Infections when a Fitness Trade-Off for Resistance is Present*

*Mentor:*Fred Adler

*Abstract:*How much does indiscriminate antibiotic use promote the spread of antibiotic-resistant infections in a population? Assuming a fitness trade-off for resistance exists, it is possible for an antibiotic-vulnerable strain to outlast a resistant type within an untreated host carrying both. That means prudent treatment schemes can potentially manage resistance levels in the population-at-large. We use a compartmental ODE model incorporating a class of co-infected individuals to analyze the epidemiology. The Next Generation Operator method gives $R_0$ for the two strains, and we perform invasion analysis for both types under different treatment schemes. Haphazard antibiotic use favors resistant strains by treating both singly-infected and co-infected people, while shrewd treatment targets those carrying the vulnerable strain only. According to the model, inattentively treating co-infected individuals significantly promotes resistance in a population.

## Session 2/2 - Monday May 4 - 3:30pm-5:30pm - Room LCB 222

03:30-03:50pm,

**Jacob Madrid**,

*Study of a Link Invariant- Stabilized Hat Heegaard Floer Homology*

*Mentor:*Sayonita Ghosh Hajra

*Abstract:*A knot can be thought of as a tangled rope with ends attached. One of the fundamental ideas in studying knots and links is determining whether two projections are equivalent. A knot invariant is a property of a knot, which is the same for equivalent knot projections. When two invariants give two different objects, then it can be said that the knots are different. However, invariants cannot be used to distinguish between knots if they result in the same object. In todays presentation, we will talk about another knot and link invariant called "Stabilized Hat Heegaard Floer Homology". This invariant is a combinatorial algorithm introduced by A. Stipsicz. Here we will present this algorithm and describe a code to change a "grid diagram" into an "extended grid diagram".

03:50-04:10pm,

**Tianyu Wang**,

*Maxima of Correlated Gaussian Random Variables*

*Mentor:*Tom Alberts

*Abstract:*We consider the maximum of correlated Gaussian random variables and examine the methods for computing the mean and distribution of the maximum. We are particularly interested in how these quantities change as a function of the correlation structure. We will focus on the 2 dimensional case.

04:10-04:30pm,

**Mackenzie Simper**,

*The Stochastic Heat Equation on Markov Chains*

*Mentor:*Tom Alberts

*Abstract:*Consider a casino with several gambling tables and a gambler who chooses to move among them randomly following the dynamics of a Markov chain. The heat equation describes the evolution of the probabilities for the gambler being at a given table at a given time. While he is at each table the gambler gains or loses a random amount of money, and the stochastic heat equation describes the evolution of his expected fortune over all possible trajectories between the tables. The fundamental solution to the stochastic heat equation is described by a matrix-valued stochastic differential equation (SDE). We explore various properties of this process of random matrices, including the evolution of the norm, the determinant, and the trace of the matrix.

04:30-04:50pm,

**Michael Zhao**,

*Spectra of Random Graph Models*

*Mentor:*Braxton Osting

*Abstract:*Given a graph $G$ from ``real-world'' data, let $G_0$ be a graph generated from a random graph model, e.g. BTER or Chung-Lu. Let $\sigma(L(H), l)$ be the vector of the smallest $l$ eigenvalues of the graph Laplacian of a graph $H$, and let $A(H)$ be the adjacency matrix. Then we consider the constrained optimization problem of adding or removing edges to $G_0$ to create a graph $G_t$ where $\sigma(L(G_t), l)$ is closer to $\sigma(L(G), l)$ than $\sigma(L(G_0), l)$, in the sense of $\ell^2$-norm, and $||A(G_0) - A(G_t)||_1 = M$ for some preset $M$. The performance of the algorithm is evaluated for a variety of random graph models, for differing values of $l$ and $M$, and also when $L(G)$ is replaced by $A(G)$.

04:50-05:10pm,

**Jeremy Allam**,

*Low-Energy Satellite Transfer from Earth to Mars*

*Mentor:*Elena Cherkaev

*Abstract:*A new type of satellite transfer that uses half the amount of fuel as conventional transfers has been discovered. This transfer, called a low-energy transfer, proved to work in 1991 when a Japanese satellite successfully went in orbit around the moon using this technique. Since then, more research has been conducted to prove that a low-energy transfer can be accomplished between the moons of Jupiter. In the case of this presentation, it is shown that a low-energy transfer is possible from Earth to Mars using similar techniques as between the Jovian moons.

05:10-05:30pm,

**Anthony Cheng**,

*Percolation Theory for Melt Ponds on Arctic Sea Ice*

*Mentor:*Kenneth Golden

*Abstract:*Extreme losses in summer Arctic sea ice pack, a leading indicator of climate change, have led to the need for significant revisions of global climate models. Part of the efforts to improve these models is an increased emphasis on sea ice albedo (reflectance), which is closely related to the formation of melt ponds on the sea ice. This project investigates the novel use of percolation theory to model melt pond formation. Innovative methods were implemented to calculate the percolation probability as a function of $p$, the probability of an edge being open. The results were analyzed to determine the critical threshold value $p_c$, which was compared with the known value of 0.50 for a two dimensional square lattice. This model was then adapted to determine the critical area threshold for melt ponds on sea ice as 0.4845, which had not been numerically determined previously. This threshold allows for simple detection of percolation, and thus also changes in the albedo and the melting rate of the sea ice. The calculation of this parameter is a first step to creating more dynamic models for the growth and disappearance of melt ponds over time.