## 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:
- A paper is submitted to 2013 LA/MS Section of MAA (Mathematical Association of America) Meeting for undergraduate student paper competition. A slightly revised version is available here. The final version is published in
- Differential Geometry - Dynamical Systems, Volume 16 (2014), 203-218.

- 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.