%Liming Wang syllabus
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\begin{document}
\title {Oral Exam Syllabus}
\author{Liming Wang}
\date{}
\maketitle

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Committee: Eduardo Sontag, Natalia Komarova, Joel Lebowitz, Yanyan Li

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\section{Major topic: Mathematical Biology}
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\subsection{Continuous models for population dynamics}
\begin{itemize}
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\item Models for single species populations: Malthus law, logistic growth
\item Predator-Prey systems and Lotka-Volterra equations
\item Populations in competition
\end{itemize}

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\subsection{Models for molecular events}
\begin{itemize}
\item Basic mass action
\item Enzyme kinetics: Michaelis-Menten, Burk plots, allosteric inhibitors, 
competitive inhibition and cooperativity (Hill equation)
\item Quasi-steady-state analysis, time-scale separation
\item Biological oscillators and switches
\item Activator-Inhibitor and positive feedback systems
\item Probabilistic channel model
\end{itemize}

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\subsection{Partial differential equation models in biology}
\begin{itemize}
\item Diffusion: the derivation of diffusion equation, speed of diffusion
\item Diffusion through a membrane: Ohm's Law
\item Facilitated diffusion
\item The membrane potential: Nernst equation
\end{itemize}

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\subsection{Excitable systems, limit cycles and oscillations}
\begin{itemize}
\item Excitable systems: Hodgkin-Huxley model, Fitzhugh's analysis of Hodgkin-Huxley equations
\item periodic orbits and limit cycles
\item Poincare-Bendixon theory
\item Hopf bifurcation
\end{itemize}

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\subsection{Development and Pattern Formation}
\begin{itemize}
\item Conditions for diffusive instability
\item Turing's approach to morphogenesis
\end{itemize}

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% PED
\section{Minor topic: Partial Differential Equations}
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\subsection{Laplace equations}
\begin{itemize}
\item Derivation of the fundamental solution
\item Mean value formulas
\item Properties of harmonic function: maximum principle, smoothness, local estimates on derivatives, Harnack inequality
\end{itemize}

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\subsection{Second order elliptic equations}
\begin{itemize}
\item Second order elliptic operator
\item Weak maximum principle
\item Hopf Lemma
\item Strong maximum principle
\end{itemize}

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\subsection{Sobolev spaces}
\begin{itemize}
\item Weak derivatives
\item Sobolev spaces
\item Approximation: interior/global approximation by smooth functions
\item Gagliardo-Nirenberg-Sobolev inequality
\item Poincare inequality
\end{itemize}

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\begin{thebibliography}{9}

\bibitem%
{J}
L. Edelstein-Keshet,
{\em Mathematical Models in Biology},
Random House, 1988;

\bibitem%
{JJ}
J. Keener and J. Sneyd,
{\em Mathematical Physiology},
Springer-Verlag, New York, Inc. 1998;

\bibitem%
{JM}
J.D. Murray,
{\em Mathematical Biology},
Springer-Verlag, 1993.

\bibitem%
{Evans}
L.C. Evans,
{\em Partial Differential Equations},
AMS Providence, 1998.

\bibitem%
{GT}
{D. Gilbarg and N.S. Trudinger,}
{\em Elliptic Partial Differential Equations of Second Order},
Springer-Verlag, 1983.

\end{thebibliography}

\end{document}
