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\begin{document}
Derek Hansen's Syllabus for the Oral Qualifying Exam
\medskip
\begin{enumerate}
  \item Functional Analysis
  \begin{enumerate}
    \item basics of Banach spaces
    \begin{enumerate}
      \item operations on Banach spaces (direct sums and quotients)
      \item linear functionals, the Hahn-Banach theorem (analytic and 
geometric forms), dual spaces
      \item the Baire Category theorem and its consequences, namely, 
the Banach-Steinhaus theorem, the open and inverse mapping theorems, and
the closed graph theorem
      \item basics of Hilbert spaces (Riesz lemma, adjoints, orthonormal 
bases, etc.)
      \item weak and weak-$*$ topologies, the Banach-Alaoglu theorem, the 
Eberlein \v{S}mulian theorem, Mazur's theorem
    \end{enumerate}



  \item measure theory on compact Hausdorff spaces (the Riesz 
representation theorem)



  \item bounded operator theory
    \begin{enumerate}
    \item adjoints
    \item resolvent set, resolvent operator and the spectrum
    \item compact operators
    \item the spectrum of a compact operator
    \item the Fredholm alternative
    \item the spectral theorem for compact normal operators on a Hilbert space
    \end{enumerate}

  \end{enumerate}




  \item Sobolev Spaces
  \begin{enumerate}
    \item $H^{s}(\rr^n)$ defined via the Fourier transform
    \item $W^{k,p}$ spaces
    \item density of $C^\infty(\bar{\Omega})$ in $W^{k,p}(\Omega)$
    \item global extension
    \item the trace operator
    \item the Gagliardo-Nirenberg-Sobolev inequality
    \item Morrey's inequality
    \item the Rellich-Kondrakov theorem
    \item Lipschitz functions and $W^{1,\infty}$
    \item Rademacher's theorem
  \end{enumerate}
 


  \item Second Order Elliptic PDE
    \begin{enumerate}
      \item Laplace's Equation
        \begin{enumerate}
         \item the fundamental solution
         \item solving Poisson's equation, $\Delta u=f$ in $\Omega$, where
$\Omega$ is a bounded domain and $f$ is locally H\"{o}lder continuous,
with the Newtonian potential of $f$
         \item mean value formula and maximum principle for subharmonic functions
         \item basic Harnack's inequality
         \item Green's function
         \item basic properties of harmonic functions (analyticity, a 
uniformly bounded family of harmonic functions has a u.c.c. convergent
subsequence)
         \item Perron's method of subharmonic functions
         \item existence of solutions to Dirichlet and Neumann problems
via layer potentials       
        \end{enumerate}
      \item weak and strong maximum principles, uniqueness theorems for 
boundary value problems
      \item definition of weak solution
      \item existence via energy estimates and the Lax-Milgram theorem
      \item existence via the Fredholm alternative
      \item eigenvalues and eigenfunctions, the Rayleigh-Ritz characterization
of the principle eigenvalue of a symmetric operator
      \item interior and boundary regularity of solutions
    \end{enumerate}
\end{enumerate}
\bigskip
References:
\begin{enumerate}
  \item Gilbarg, David and Trudinger, Neil S. {\em Elliptic 
Partial Differential Equations of Second Order}
  \item Evans, Lawrence. {\em Partial Differential Equations}
  \item Folland, Gerald B. {\em Introduction to Partial Differential Equations}
  \item Br\'{e}zis, Ha\"{\i}m. {\em Analyse Fonctionelle}
  \item Reed, Michael and Simon, Barry. {\em Functional Analysis}
  \item Protter, Murray H. and Weinberger, Hans F. {\em Maximum 
Principles in Differential Equations}
\end{enumerate}
\end{document}

