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:
      • 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.

    • 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.