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\begin{center} \Large Oral Qualifying Exam Syllabus \\
Yongzhong Xu \\
December 1, 2000\\ 
\end{center}
\medskip


\begin{center}  \large 
\textbf{Major Topic: Partial Differential Equations}
\end{center}

\begin{enumerate}
\item \textbf{Laplace's Equation:} fundamental solutions, mean value
formulas, properties of harmonic functions, Green's function,
Harnack's inequality, Perron's methods

\item \textbf{Sobolev Spaces: }
\begin{enumerate}
\item weak derivatives:  definition of Sobolev spaces
\item approximations: interior/global approximation by smooth functions 
\item Extensions, traces, compactness
\item Sobolev inequalities:  Gagliado-Nirenberg-Sobolev inequality, 
Morrey's inequality, general Sobolev inequalities, Poincar\'e inequalities 
\item Fourier transformation methods
\end{enumerate}

\item \textbf{Second order linear elliptic equations: }
\begin{enumerate}
\item Existence of weak solutions: Lax-Milgram theorem, energy
  estimates, Fredholm alternatives
\item Schauder estimate
\item Maximum Principles: weak maximum principle, strong maximum
  principle, Hopf's lemma, Harnack's inequality
\end{enumerate}

\item \textbf{The calculus of Variations:}
\begin{enumerate}
\item First variation, Euler-Lagrange equation
\item Existence of minimizers:  coercity, lower semicontinuous, 
convexity, weak solutions of Euler-Lagrange equeations
\item Morse theorem, Mountain Pass theorem 
\end{enumerate}

\item \textbf{Nonvariational Techniques: }
\begin{enumerate}
\item Monotonicity methods
\item Fixed point methods:  Banach's fixed point theorem, Schaulder's 
and Schaefer's fixed point theorems
\item Methods of subsolutions and supersolutions
\end{enumerate}

\end{enumerate}

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\begin{center}  \large \textbf{Minor Topic: Algebraic Topology}
\end{center}
\begin{enumerate}
\item Homotopy theory: covering spaces, lifting
criterion, higher homotopy groups, van Kampen Theorem
\item Singular homology theory: definitions, 
homotopy invariance, relative homology, excision theorem,
Mayer-Vietoris sequences, Betti numbers and Euler characteristics
\item Orientation and dua;ity on manifolds:  orientation of a manifold,
singular cohomology, cup and cap product, Poincar\'e duality 
\item Fibrations: definition, homotopy property, an example--loop spaces
\item CW-complexes: definition and the homology calculation
\end{enumerate}

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