Student Talks

Graduate Student Talks - Kirby 120

11 – 11:15 Colin Martin – A brief introduction to Hopf Algebras, Dr. Frank Moore

Abstract: Hopf algebras have held sustained academic interest since the middle of the 20th century due to their natural occurrence in quantum mechanics, topology, combinatorics and modern algebra itself. Despite recent interest, most introductory texts tend to be terse, exclusively categorical and poorly motivated. This presentation targets students with an undergraduate understanding of modern algebra. We start by defining the tensor algebra of a group module and work towards a abstraction of groups that is satisfying from a representation theoretic perspective: a Hopf algebra.

11:15 – 11:30 Aaron Rapp – Dual-Wind Discontinuous Galerkin Method with Applications to Obstacle Problems, Advisors: Dr. Tom Lewis and Dr. Yi Zhang

Abstract: A discontinuous Galerkin (DG) finite-element interior calculus has been developed as a common framework to describe various DG approximation methods for second-order elliptic problems. In this presentation, we will discuss the dual-wind DG method and its application to the obstacle problem with Dirichlet boundary conditions. In particular, we will consider applying the method to the problem −Δu≥f on Ω with u=g on ∂Ω and u≥ψ on Ω, where ψ is the given obstacle. We will also present the results for numerical tests that validate the performance of the method.

11:30 – 11:45 Katrina Lu – "Asymptotic Boundary Observability For The Wave Equation On Simplicies", Hans Christianson

Abstract: We consider the wave equation on an n-dimensional simplex with Dirichlet boundary conditions. Our main result is an asymptotic observability identity from any one side. The proof is approached by using commutator and integration by parts arguments.

11:45 – 12 Max Razek - Why are Leaves Wavy?, John Gemmer

Abstract: In the non-Euclidean model of elasticity, growth is modeled by a Riemannian metric that encodes local changes in distance. In response to the growth, the sheet deforms to minimize an elastic energy. The elastic energy consists of the sum of the stretching and bending energy. Minimizers of the stretching energy consist of isometric immersions of the metric, while minimizers of the bending energy remain flat. The competition between bending and stretching selects a pattern in the sheet. In this talk, we will show that periodic patterns have the lowest energy for a large class of metrics. Qualitatively, our results agree with patterns observed in leaves and torn elastic sheets.

Graduate Student Talks - Manchester 121

11 – 11:15 Susan Rogowski – Modeling DNA Replication Using the sine-Gordon Equation, Dr. Sarah Raynor

Abstract: DNA replication begins when local unwound regions of several broken hydrogen bonds form. These regions are often referred to as bubbles and their formation can be modeled using a chain of coupled pendulums. The motions of angular oscillations that occur are usually modeled using the sine-Gordon equation. In this model, the discrete analog of the sine-Gordon equation is derived. Instead of taking the continuous limit, the space step between pendulums remains nonzero and the forces on the pendulum by Newton’s laws of motion are considered. Additionally, the randomization of nitrogen bases, Adenine, Thymine, Guanine, and Cytosine, within the chain of pendulums is modeled by randomly selecting the corresponding coefficients for each base. The system of differential equations is solved and plotted using Verlet integration in MATLAB.

11:15 – 11:30 Kristen Scheckelhoff/Ayesha Ejaz – Balancing the cost of infection: The effect of clean needle use on the spread of hepatitis C among injecting drug users. [Co-presenter: Ayesha Ejaz, UNCG; Advisor: Dr. Igor Erovenko, UNCG]

Abstract: Hepatitis C is an infectious liver disease which contributes to an estimated 400,000 deaths each year. The disease is caused by the hepatitis C virus (HCV) and is spread by direct blood contact between infected and susceptible individuals. Despite the magnitude of its impact on human populations, hepatitis C receives relatively little scientific attention. In particular, studies targeting disease eradication in the populations most at risk -- injecting drug users -- are scarce. Here we construct a game-theoretic model to investigate the effect of clean needle use on the spread of HCV. Individual drug users, seeking to maximize their own payoff by weighing the cost of clean equipment against the cost of infection, may opt for a level of protection between 0 and 100%. We find that the spread of HCV in this population can theoretically be eliminated if individuals use sterile equipment approximately two-thirds of the time. Achieving this level of compliance, however, requires that the real and perceived costs of obtaining sterile equipment are essentially zero.

11:30 – 11:45 Difan Li – Local Energy Estimates for Wave Equations with Degenerate Trappings; Advised by Dr. Jason Metcalfe

Abstract: My project focuses on studying local energy estimates for wave equations, especially for wave equations on background geometries where trapping occurs. In particular, I will take a close look into a previously unexamined scenario where the trapping comes from an inflection point in the generating function of a surface of revolution. My project studies a warped product manifold whose generating function has an inflection point, which corresponds to trapped rays. Based on the preceding work of Christianson and Metcalfe on local smoothing estimates for the Schrödinger equation on geometries with inflection transmission, which is an analog of local energy estimates for wave equations on geometries with an inflection point, it is expected that the resulting local energy estimates will necessitate an algebraic loss of smoothness, and this will be only the second explicit example of this phenomenon for wave equations. This will be a fundamental contribution to the understanding of the behavior of wave equations on geometries with trapping.

11:45 – 12 Ivanti Galloway – False Discovery Rate Analysis, Dr. Jan Rychtář

Abstract: In October 2015, a New York Times article highlighted a disparity between the proportion of black versus non-black drivers pulled over in traffic stops in Greensboro, NC. In response to these allegations, we examined 563 individual officers in the Greensboro Police Department (GPD) to determine if bias against blacks played a role in their traffic stops. We used propensity score weighting, which compared an officer’s particular stops to similar stops made by peers. This method was based on RAND Corporation’s study for the Cincinnati Police Department. For our purposes, two stops were similar if they occurred for the same reason at a similar time of day and at a similar location in town. After applying our propensity score weights, we conducted a false discovery rate (fdr) analysis. In this analysis, 10 out of the 563 officers had z-statistics that indicated racial bias against black drivers. These results are based off of 295,228 stops that occurred between January 1, 2009 and September 30, 2015.

Undergraduate Student Talks - Manchester 124

11 - 11:15 Addie Harrison - Human Visual Tracking and Image Formation Advisor: John Gemmer

Abstract: Animals in motion have the ability to track an object with their vestibular ocular system in order to keep the image stationary on the retina. A motivating example is a chicken’s ability to keep their head stationary despite the forces that are being applied to their body. In our work, we are interested in understanding the mathematics of image formation and image tracking. Using tools from vector calculus, projective geometry and linear algebra, we developed a model that maps images from space to the retina. This model allowed us to create optical flow vector fields that shows the path light takes on the retina. Furthermore, using differential equations, we developed a dynamic 3D model of the head coupled to the eyes which track an object in space. In collaboration with neuroscientists we can use our models to learn how the visual system and the brain are used in conjunction to stabilize an image as well as to process depth perception.

11:15 – 11:30 Brady Gales - The Mathematics of the Head Injury Criterion, Advisor: Dr. John Gemmer

Abstract: The Head Injury Criterion (HIC) score is used as a predictor for the outcome of a head injury during front impact collision and is widely used in North American motor vehicle safety regulations, the assessment of playground safety, and in the assessment of athletic helmets. In particular, the HIC score measures local rapid fluctuations in the acceleration curve during an accident to predict whether a serious injury will occur. However, the HIC score as it is currently formulated is ill-posed in the sense that acceleration curves that minimize the HIC score is a discontinuous function. This result is physically unsatisfactory as it leads to infinite jerk (divergent third derivative). Our research consists of studying a regularized version of this functional that penalizes sharp deviations in the jerk. We reformulated the HIC score in a manner that smooths out the divergent third derivative and mathematically proved that it is possible to minimize this modified version of the HIC score.

11:30 – 11:45 Jessica Stevens - A Mathematical Model for Tumor Growth and Treatment Using Virotherapy, Dr. Zach Abernathy and Dr. Kristen Abernathy

Abstract: We present a system of four nonlinear differential equations to model the use of virotherapy as a treatment for cancer. This model describes interactions among infected tumor cells, uninfected tumor cells, effector T-cells, and virions. Using various stability analysis techniques, we establish a necessary and sufficient treatment condition to ensure a globally stable cure state. We additionally show the existence of a cancer persistence state when this condition is violated and provide numerical evidence of a Hopf bifurcation under estimated parameter values from the literature. We conclude with a discussion on the biological implications of our results.

11:45 – 12 Dave Segall - "Promoting Effective Teaching Through Comprehensive Evaluation Systems" with Dr. Dalzell

Abstract: Teachers live at the forefront of education; they are the ones in classrooms with students each and every day. As such, a quality teaching workforce is imperative to have a successful educational system—whether it be a single classroom, a school, or an entire district. But how do we know what a quality teaching workforce looks like, and how can we ensure students get the quality teachers they deserve? In my talk, I address the idea behind this question by examining the role teacher evaluation has in creating better teachers. To do so, we must first understand what it means to be an ‘effective teacher,’ so I explore the theory and research underlying our ideas of quality teaching as well as methods for assessing teachers. Secondly, I examine the implementation of a specific teacher evaluation model currently used in Washington, D.C. to lay the framework for a comprehensive system. Lastly, I apply the theory of effective teaching and its evaluation to a local context: the teacher evaluation model used in North Carolina public schools. Ultimately, I analyze teacher evaluation data to showcase the potential impact a comprehensive evaluation system can have on teacher effectiveness and the ways administrators can use such systems to benefit both their teachers and their students.