 ## Research with Undergraduate Students

Research is an important part of mathematics and physics education. I maintain a program of research projects in mathematics, theoretical physics, theoretical computer science, mathematical biology and mathematical finance that a bright and enthusiastic undergraduate student can tackle. Those projects are relevant to current research. For students, research is both a learning process and a discovery process.

If you are interested in undergraduate research, please contact me by e-mail or come by my office for more details.

• ### Current Undergraduate Research

• #### Elementary Qubit Gates of PT-Symmetric Quantum Computing

• Status: Current
• Student: Andrew Hood, University of Southern Mississippi
• Project Synopsis:
• In this project, we study elementary qubit gates of PT-symmetric quantum computing.

• ### Past Undergraduate Research

• #### Surfaces of Revolution with Constant Mean Curvature $$H=c$$ in Hyperbolic 3-Space $$\mathbb H^3(-c^2)$$

• Status: Complete
• Student: KinseyAnn Zarske, University of Southern Mississippi
• Abstract: We construct surfaces of revolution with constant mean curvature $$H=c$$ in hyperbolic 3-space $$\mathbb H^3(-c^2)$$ of constant curvature $$-c^2$$. It is shown that the limit of the surfaces of revolution with $$H=c$$ in $$\mathbb H^3(-c^2)$$ is catenoid, the minimal surface of revolution in Euclidean 3-space as $$c$$ approaches $$0$$.
• Publication:
• Talks:
• Student Presentations, 2013 LA/MS Section of MAA Meeting. Her presentation in pdf format is available here.
• Animations: I made some animations in regard to this project.
• Animation 1: Animation of profiles curves for CMC $$H=c$$ surfaces of revolution in $$\mathbb H^3(-c^2)$$ tending toward the profile curve of catenoid in $$\mathbb E^3$$ as $$c\to 0$$.
• Animation 2: Animation of CMC $$H=c$$ surfaces of revolution in $$\mathbb H^3(-c^2)$$ tending toward catenoid in $$\mathbb E^3$$ as $$c\to 0$$.
• Animation 3: Animation 2 with catenoid in $$\mathbb E^3$$.
• Animation 4: Animation of minimal surfaces of revolution in $$\mathbb H^3(-c^2)$$ tending toward catenoid in $$\mathbb E^3$$ as $$c\to 0$$.
• #### Quantum Calculus

• Status: Incomplete
• Student: Lawrence Warren, Alcorn State University, Summer AGEM REU 2012
• #### Shape of Sound

• Status: Complete
• Student: Jarred Jones, Jackson State University, Summer AGEM REU 2011
• Abstract: In this project, we model a vibrating drumhead. A vibrating drumhead can be modeled by the wave equation in polar coordinates. The poster of this project can be viewed here and the animation of a vibrating drumhead can be viewed here.
• Project Report: A report from this research project is available in pdf format here.
• #### Timelike Constant Mean Curvature Surfaces of Revolution in Minkowski 3-Space

• Status: Complete
• Student: Jeffrey H Varnado, University of Southern Mississippi, 2007
• Abstract: First, we study certain ODEs that characterize timelike surfaces of revolution with constant mean curvature in Minkowski 3-space. These ODEs are non-linear and it is very difficult to find their solutions explicitly. Numerical solutions to these ODEs can be found by well-known numerical methods such as Runge-Kutta’s or Euler’s methods. We obtain examples of such surfaces from the numerical solutions.
• Publication: Differential Geometry and Dynamical Systems 9, No. 1 (2007), 82-102.
• #### Spacelike Constant Mean Curvature Surfaces of Revolution in Minkowski 3-Space

• Status: Complete
• Student: Jeffrey H Varnado, University of Southern Mississippi, 2006
• Abstract: This paper studies various ordinary differential equations that characterize spacelike constant mean curvature surfaces of revolution in Minkowski 3-space. Those differential equations are nonlinear and cannot be solved explicitly. Using numerical methods such as Runge-Kutta’s or Euler’s methods, we solve those differential equations and obtain examples of spacelike constant mean curvature surfaces of revolution in Minkowski 3-space.
• Publication: Differential Geometry and Dynamical Systems 8, No. 1 (2006), 144-165.