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.

• #### Mathematical Modeling of Epidemics

• Status: Current
• Student: Rachel Prather, Master's in Mathematics, University of Southern Mississippi
• Project Synopsis:
• 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.

• #### 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.