\documentclass{article}

% ----------------------------------------------------------------
\begin{document}

\title{Syllabus for the oral exam of Pablo Angulo}%
\author{Pablo Angulo}%
\date{4-26-05}%

% ----------------------------------------------------------------
\maketitle

% ----------------------------------------------------------------
\section{Elliptic linear PDE}
\textbf{1-Sobolev spaces}
\begin{itemize}
  \item Definition of Sobolev and Holder spaces.
  \item Basic properties, traces and extensions.
  \item Embeddings of Sobolev and Holder spaces.
  \item Compact embeddings.
\end{itemize}
\textbf{2-Linear elliptic second order operators}
\begin{itemize}
  \item Existence: Lax Milgram and Fredholm theorems.
  \item Regularity of solutions.
  \item Maximum principles.
  \item Eigenvalues for elliptic operators.
\end{itemize}
\textbf{3-Elliptic operators on manifolds}
\begin{itemize}
  \item Definition of differential operator. Symbol of an operator.
  \item Pseudodifferential operators.
  \item Existence of a pseudo-inverse for differential operators.
  \item Elliptic complexes. Hodge theory. Poincare duality.
\end{itemize}

\section{Differential topology}
\textbf{1-Foundations}
\begin{itemize}
  \item Differentiable Manifolds.
  \item Transversality.
  \item Critical points of smooth maps. Sard's theorem. Morse functions.
  \item Orientations and Integration on Manifolds.
\end{itemize}
\textbf{2-Brouwer degree and intersection numbers}
\begin{itemize}
  \item Degree of a continuous map.
  \item Applications of degree. Bouwer fixed point, invariance of domain.
  \item Intersection numbers. Euler number of a vector bundle.
\end{itemize}
\textbf{3-De Rham cohomology}
\begin{itemize}
  \item Poincare lemma.
  \item Mayer-Vietoris sequence.
  \item Poincare dual of a closed oriented manifold.
\end{itemize}
\textbf{4-Morse theory}
\begin{itemize}
  \item Morse lemma.
  \item The Morse inequalities.
  \item Morse Homology. Poincare duality.
\end{itemize}
% ----------------------------------------------------------------
\end{document}

