% Oral Quals for Spaceghost, May 3, 2005
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\title{Oral Qual Syllabus:\\Set Theory; Ergodic Theory}
\author{Samuel Coskey}
\begin{document}
\maketitle

\subsection*{Set Theory}

\begin{itemize}

\item Basic set theory (Jech 1-9)
\begin{itemize}
 \item K\"onigs lemma, cardinal arithmetic in ZFC
 \item Fodor's lemma, Solovay's theorem
 \item Ramsey's theorem
 \item there exists an Aronsajn tree
 \item Shanin's root system lemma ($\times 2$)
\end{itemize}

\item Descriptive set theory (Jech 11)
\begin{itemize}
 \item elementary properties of Polish spaces
 \item non-collapsing of the Borel hierarchy
\end{itemize}

\item Models of set theory (Jech 12,13)
\begin{itemize}
 \item Mostowski's collapsing theorem
 \item relativization and absolute formulas
 \item reflection principle
 \item $L$ is a model of ZFC, $L_\alpha$ is absolute (statements)
 \item condensation lemma
 \item $L$ is a model of GCH, $\diamondsuit$
\end{itemize}

\item Forcing basics (Kunen, ch. VII, VIII)
\begin{itemize}
 \item fundamental forcing theorems (statements)
 \item chain and closure conditions on a partial order
 \item mixing and maximal lemmas
 \item Cohen and L\'evy forcing
 \item product and iterated forcing
 \item factor lemma (statement)
 \item independence of MA$+\neg$CH
 \item consequences of MA, e.g. $2^\omega$ is regular
 \item Easton supported forcing, Easton's theorem
\end{itemize}

\item Large cardinals (Jech 17, Kanamori 5, Joel's notes)
\begin{itemize}
 \item measurable cardinals and embeddings $j:V\rightarrow M$
 \item L\'evy-Solovay theorem (small forcing and measurables)
 \item lifting criterion; diagonalization criterion
 \item find a weakly compact cardinal in $L$
 \item supercompact cardinals have a Laver function
 \item Silver's theorem (failure of GCH at a measurable)
 \item Laver's theorem (the Laver preparation)
 \item Reinhardt cardinals
\end{itemize}

\end{itemize}

\newpage

\subsection*{Ergodic Theory}

\begin{itemize}

\item Real analysis (Walters, others)
\begin{itemize}
 \item regularity properties of Borel measures
 \item Jensen's inequality
 \item Riesz representation theorem (statement)
 \item spectral theorem for unitary operators (statement)
 \item conditional expectation
 \item existence and properties of Haar measure (statements)
 \item characters of compact abelian groups (statements)
\end{itemize}

\item Measure-preserving transformations (Walters 1)
\begin{itemize}
 \item recurrence theorem
 \item equivalent formulations of ergodicity
 \item equivalent formulations of weak and strong mixing
 \item Von-Neumann mean ergodic theorem
 \item Garsia maximal inequality
 \item Hopf ergodic theorem
 \item Birkhoff ergodic theorem
\end{itemize}

\item Examples
\begin{itemize}
 \item ergodic transformation on $\mathbb R$
 \item group rotations, endomorphisms, and affine transformations
 \item Bernoulli shifts
\end{itemize}

\item Discrete and continuous spectrum (Walters 2,3)
\begin{itemize}
 \item notions of similarity
 \item multiplication theorem
 \item spectrum of an ergodic transformation
 \item continuous spectrum theorem
 \item discrete spectrum theorem
 \item representation theorem for discrete spectrum
\end{itemize}

\item Topological dynamics (Walters 5.2-4)
\begin{itemize}
 \item minimality
 \item transitivity
\end{itemize}

\item Unique ergodicity (Walters 6.5)
\begin{itemize}
 \item the topological space of invariant measures
 \item existence of ergodic measures
 \item characterizations of unique ergodicity
 \item basic facts about $g$-measures
\end{itemize}


\end{itemize}


\subsection*{References}
\begin{itemize}
\item Jech, Thomas.  Set Theory.  \emph{Third} Edition.
\item Kanamori, Akihiro.  The Higher Infinite.
\item Kunen, Kenneth.  Set Theory: An Introduction to Independence Proofs.
\vspace {10pt}
\item Halmos, Paul R.  Lectures on Ergodic Theory.
\item Walters, Peter.  An Introduction to Ergodic Theory.
\end{itemize}


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