## 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

#### Mathematical Modeling of Epidemics

- Status: Current
- Student: Rachel Prather, Master's in Mathematics, University of Southern Mississippi
- Project Synopsis:
### Past Graduate Students

#### 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 review some of the known mathematical models of edpidemics including stochastic models and explore the possibility of developing an improved model of epidemics.