## Research with Graduate Students

If you are a master's student and wish to do research in mathematics, theoretical physics, theoretical computer science, mathematical biology, or mathematical finance for your master's thesis, please contact me by e-mail or come by my office for more details.

### Current Graduate Students

#### Applied Geometrical Methods

- Status: Current
- Student: Robert Hulsey, Ph.D. in Computational Sciences with Emphasis in Mathematics, University of Southern Mississippi
- Project Synopsis:
#### Mathematical Modellings in Biological Sciences

- Status: Current
- Student: Rachel Prather, Ph.D. in Computational Sciences with Emphasis in Mathematics, University of Southern Mississippi
- Project Synopsis:
### Past Graduate Students

#### On a Stochastic Model of Epidemics

- Status: Complete
- Student: Rachel Ann Prather, Master's in Mathematics, University of Southern Mississippi, 2021
- Abstract: This thesis examines a stochastic model of epidemics initially proposed and studied by Norman T. J. Bailey [1]. We discuss some issues with Bailey’s stochastic model and argue that it may not be a viable theoretical platform for a more general epidemic model. A possible alternative approach to the solution of Bailey’s stochastic model and stochastic modeling is proposed as well. Regrettably, any further study on those proposals will have to be discussed elsewhere due to a time constraint.
- Publication: Thesis is available in pdf format here.
#### Lorentz Invariant Spacelike Surfaces of Constant Mean Curvature in Anti-de Sitter 3-Space

- Status: Complete
- Student: Jamie Patrick Lambert, Master's in Mathematics, University of Southern Mississippi, 2015
- Abstract:In this thesis, Jamie contruct Lorentz invariant spacelike surfaces of constant mean curvature \(H=c\) and maximal Lorentz invariant spacelike surfaces in anti-de Sitter 3-space \(\mathbb{H}^3_1(-c^2)\). He also studied the limit behavior of those surfaces. It turns out that as \(c\to 0\) Lorentz invariant spacelike surfaces of constant mean curvature \(H=c\) and maximal Lorentz invariant spacelike surfaces in \(\mathbb{H}^3_1(-c^2)\) both approach the maximal spacelike catenoid in Minkowski 3-space \(\mathbb{E}^3_1\).
- Publication: Thesis is available in pdf format here.
- Animations: I made some animations in regard to this project.
- Animation 1: Animation of Lorentz invariant spacelike surfaces of CMC \(H=c\) in \(\mathbb{H}^3_1(-c^2)\) that approach the maximal spacelike catenoid in \(\mathbb{E}^3_1\) as \(c\to 0\).
- Animation 2: Animation of maximal Lorentz invariant spacelike surfaces in \(\mathbb{H}^3(-c^2)\) that approach catenoid in \(\mathbb{E}^3_1\) as \(c\to 0\).
#### Quantum Mechanics as a Gauge Theory

- Status: Complete
- Student: Joseph (Joey) L. Emfinger, Master's in Mathematics, University of Southern Mississippi, 2010
- Abstract: It is well knwon that quantum mechanics can be treated as a gauge theory by considering quantum state functions as sections of a complex vector bundle over Minkowski spacetime. In this thesis, we propose an alternative approach to a gauge theoretic treatment of quantum mechanics. A quantum state function \(\psi:\mathbb{R}^{3+1}\longrightarrow\mathbb{C}\) can be lifted to a map (called a lifted state) to the holomorphic tangent bundle \(T^+(\mathbb{C})\), where we regard \(\mathbb{C}\) as a Hermitian manifold. The map can be regarded as a holomorphic section (a vector field) of \(T^+(\mathbb C)\) parametrized by space-time coordinates. The probability density of a lifted state function is naturally defined by Hermitian metric on \(\mathbb C\). It turns out that the probability density of a state function coincides with that of its lifted state. Furthermore, the Hilbert space structure of state functions is solely determined by the Hermitan structure defined on each fibre \(T_p^+(\mathbb C)\) of \(T^+(\mathbb C)\). This means that physically a state and its lifted state are not distinguishable and we may study a quantum mechanical model with lifted states in terms of differential geometry, consistently with the standard quantum mechanics. In particular, we discuss quantum mechanics of a charged particle in an electromagnetic field as an abelian gauge theory.
- Publication: Thesis is available in pdf format here. A paper out of his thesis is published in Synergy, Volume 3, Issue 2, Summer 2012. Synergy is a Journal for Graudate Student Research published by the Graduate School at the University of Southern Mississippi.

In this project, we consider geometric modellings of problems in physics, computer science, and finance, and study their solutions.

In this project, we consider mathematical modellings (deterministic or stochastic) of problems in biological sciences and study their solutions.