\documentclass[12pt]{article}
\usepackage{enumerate}

\begin{document}

\centerline{\huge Syllabus}

\bigskip

\centerline{\large Committee: Yanyan Li, Sagun Chanillo, Zheng-Chao Han, Feng Luo}

\begin{enumerate}[I.]
\item {\bf Major Topic: Partial Differential Equations}
\begin{enumerate}[1.]
\item {\bf First order partial differential equations: Wave-particle duality}
\begin{itemize}
\item Space of $1$-jets
\item Standard contact structure and its associated symplectic form in the space of $1$-jets
\item Characteristic direction
\item Reduction of a nonlinear first order PDE to a system of nonlinear fisrt order ODE's
\item Local existence
\end{itemize}
\item {\bf Second order elliptic equations}
\begin{enumerate}[(a)]
\item Constant coefficient case: Laplace equation
\begin{itemize}
\item Mean value formula
\item Maximum principle
\item Uniqueness
\item Regularity
\item Harnack inequality
\item Representation formulas
\end{itemize}
\item Sobolev spaces
\begin{itemize}
\item Completeness, reflexivity, density, extensions
\item Gagliardo-Nirenberg-Sobolev inequalities
\item Morrey inequality
\item Poincar\'{e} inequality
\item Rellich-Kondrachov compact embedding theorem
\item Border line case: $W_0^{1,n}$ $\hookrightarrow$ $?$
\end{itemize}
\item Weak solutions: $L^2$ theory
\begin{itemize}
\item Lax-Milgram theorem and Fredholm alternative
\item Existence and uniqueness
\item Regularity
\item Maximum principle
\end{itemize}
\end{enumerate}
\item {\bf Second order elliptic systems: Minimizers of quadratic functionals}
\begin{enumerate}[(a)]
\item Convexity
\begin{itemize}
\item Elliptic condition and convexity
\item Legendre-Hadamard condition and quasi-convexity
\end{itemize}
\item Existence
\begin{itemize}
\item Coercivity
\item Weak lower-semicontinuity
\end{itemize}
\item Regularity in 2-d: Morrey theorem
\item Regularity for linear elliptic systems
\item Regularity for continuous minimizers
\end{enumerate}
\end{enumerate}
\item {\bf Minor Topic: Riemannian Geometry}
\begin{enumerate}[1.]
\item Differentiable manifolds
\begin{itemize}
\item Tangent and cotangent bundles
\item Vector fields and forms
\item Bracket
\item Immersions and embeddings
\end{itemize}
\item Riemannian metrics
\begin{itemize}
\item Length
\item Volume
\end{itemize}
\item Connections
\begin{itemize}
\item Affine connections
\item Riemannian connections
\item Covariant derivative along a curve
\end{itemize}
\item Geodesics
\begin{itemize}
\item Geodesic equation
\item Geodesic flow
\item Exponential map
\item Gauss lemma
\item Minimizing properties
\end{itemize}
\item Curvatures
\begin{itemize}
\item Curvature
\item Bianchi identities
\item Sectional curvature
\item Ricci curvature
\item Scalar curvature
\end{itemize}
\item Jacobi fields
\begin{itemize}
\item Jacobi equation
\item Conjugate points
\end{itemize}
\item Completeness
\begin{itemize}
\item Hopf-Rinow theorem
\item Hadamard theorem
\end{itemize}
\end{enumerate}
\end{enumerate}

\begin{thebibliography}{}
\bibitem[Ar]{Ar} Vladimir I. Arnold, {\it Lectures on Partial Differential Equations}, Springer, 2004.
\bibitem[C]{C} Manfredo P. Do Carmo, {\it Riemannian Geometry}, Birk\"{a}user, 1992.
\bibitem[E]{E} Lawrence C. Evans, {\it Partial Differential Equations}, AMS, 1998.
\bibitem[Gi]{Gi} Enrico Giusti, {\it Direct Methods in the Calculus of Variations}, World Scientific, 2003.
\bibitem[GT]{GT} David Gilbarg, Neil S. Trudinger, {\it Elliptic Partial Differential Equations of Second Order}, Springer, 2001.
\bibitem[M]{M} Charles B. Morrey, Jr.,  {\it Multiple Integrals in the Calculus of Variations}, Springer, 1966.
\end{thebibliography}
\end{document}
