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\title{Oral Qualifying Exam Syllabus \\ Descriptive Set Theory; Classical Groups}

\author{Scott Schneider}

\begin{document}

\maketitle

\noindent \textbf{Classical Descriptive Set Theory}

\vspace{.05in}

\begin{itemize} \item Basic Set Theory (Jech 1-8). \end{itemize}

\vspace{.05in}

\noindent Polish and Standard Borel Spaces.

\begin{itemize}

\item Basic definitions and examples: $\mathcal{N}$,
$\mathcal{C}$, $\mathbb H$, $\mathbb R$, $[0,1]$,
$[0,1]\setminus\mathbb Q$; embeddings between them; topological
properties; $\mathbb Q$ not Polish.

\item A topological space is Polish if and only if it is
homeomorphic to a $G_\delta$ subset of $\mathbb H$.

\item Every Borel subset of a Polish space is a continuous image
of $\mathcal{N}$ and a continuous, injective image of a closed
subset of $\mathcal{N}$.

\item The Borel Isomorphism Theorem.

\item Borel-generated topologies and the Ramsey-Mackey Theorem.

\item Sequential trees, including the rank function and
Kleene-Brouwer ordering; systems of sets and their associated
maps; Souslin, Lusin, and Cantor schemes; the Souslin operation
$\mathcal{A}$.

\item Cantor-Bendixson analysis on Polish spaces.

\end{itemize}

\vspace{.1in}

\noindent The Borel and Projective Hierarchies.

\begin{itemize}
\item Basic definitions and facts, including closure properties.

\item Existence of universal sets for each; non-collapsing of
each. There does not exist a universal Borel set; there does exist
a universal analytic set.

\item Every uncountable Polish space contains an analytic set that
is not Borel.

\item The reduction theorem for additive Borel classes and the
separation theorem for multiplicative Borel classes.

\item Equivalence of various definitions of analytic sets.

\item Every coanalytic set is a union of $\aleph_1$ Borel sets.

\item Definitions of $\Sigma_1^1$-complete, $\Pi_1^1$-complete; WF
is $\Pi_1^1$-complete.

\item Regularity properties: every analytic subset of a Polish
space is measurable, has the Baire property, and has the perfect
set property.

\item Strong measure zero and Lusin sets.

\item Souslin's Theorem and the First Separation Theorem for
analytic sets.

\end{itemize}

\newpage

\noindent \textbf{Geometry of the Classical Groups}.

\vspace{.1in}

\noindent Chapters 1-5, 7-8 of Taylor.

\begin{itemize}

\item Group Actions: primitive groups; Iwasawa's Theorem.

\item Affine and Projective Geometry: basic definitions and the
fundamental theorems.

\item The groups $\Gamma L(V)$, $GL(V)$, $SL(V)$, $P\Gamma L(V)$,
$PGL(V)$, and $PSL(V)$.

\item Transvections in $SL(V)$.

\item  The simplicity of $PSL(V)$, using Iwasawa.

\item The BN-pair and split BN-pair axioms.

\item Flags, chambers, apartments, and buildings.

\item The BN-pair of $SL(V)$, including the Weyl group.

\item Correlations of $\mathcal{P}(V)$, $\sigma$-semilinear
isomorphisms  $V\rightarrow V^*$, and non-degenerate
$\sigma$--sesquilinear forms on $V$.

\item The Birkhoff-von Neumann theorem on the classification of
forms.

\item Symmetric, alternating, hermitian, and quadratic forms.

\item Witt's theorem.

\item Bases of orthogonal hyperbolic pairs.

\item The group $\Gamma L^*(V)$.

\item Polar frames and the building of a polarity.

\item The groups $Sp(V)$ and $\Gamma Sp(V)$.

\item Symplectic bases.

\item Symplectic transvections.

\item The simplicity of $PSp(V)$, using Iwasawa.

\item Symplectic BN-pairs and symplectic buildings.

\end{itemize}

\vspace{.5in}

\noindent \textbf{References}

\vspace{.05in}

\begin{itemize}

\item Jech, Thomas. \emph{Set Theory} (Third Edition).

\item Srivastava, S.M. \emph{A Course on Borel Sets}.

\item Taylor, Donald. \emph{The Geometry of the Classical Groups}.

\end{itemize}

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