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\title{Stability Theory and Algebraic Geometry \\ Oral Qualifying Exam Syllabus}
\author{Jeffrey Burdges}
\maketitle


\section{Model Theory}
\begin{itemize}
\item  Completeness and Compactness Theorems \cite[Sec. 2.1]{CK}
\item  Omitting Types and Beth's Definability Theorems \cite[Sec. 2.2]{CK}
\item  Atomic and Prime Models \cite[Sec. 2.3]{CK}
\item  $\omega$-categorical theories \cite[Sec. 2.3]{CK}  %%% Ho
\item  Quantifier Elimination for ACF, RCF, and DCF \cite[I \S2,II \S2]{MMP}
       %% \cite[p. 132]{Bo}
       %% Nullstelinstatz and Hilbert's 17th problem (2.9  p. 12)
\item  Saturated Models \cite[Sec. 2.3]{CK}  %%% Ho
\end{itemize}


\section{Stability Theory}
\begin{itemize}
\item  Strongly Minimal Sets \cite[Ch. 9]{Ho}
\item  Definability of Types, Heirs, and Forking \cite[Ch. 1,2,3,4]{Pi1}
\item  Open Mapping and Finite Equivalence Relation Theorems \cite[Ch. 4]{Pi1}
\item  Ranks \cite[Ch. 6]{Pi1}
\item  Indiscernibles \cite[Ch. 7]{Pi1}
\item  Somewhat Saturated Prime Models \cite[Ch. 8]{Pi1}
\item  Morley's Theorem \cite{Ho,Pi1} %%% \cite[Sec. 7.1]{CK}
\end{itemize}
  

\section{Geometric Stability Theory}
\begin{itemize}
\item  Imaginary Elements and Elimination of Imaginaries \cite[Ch. 1]{Pi2}
\item  Canonical Parameters and Canonical Base \cite[Ch. 1]{Pi2}
\item  Orthogonality \cite[Ch. 1, Ch. 2]{Pi2}
       %% Assumption 5.7, Lemma 5.12 (p. 55)
% \item  Minimal Types and 1-Based Theories \cite[Ch. 2]{Pi2}
% \item  Dichotomy Theorem and Zilber's Theorem \cite[Ch. 2]{Pi2}
% \item  Group Configuration Theorem \cite[Ch. 5]{Pi2}
\end{itemize}

  
\section{Groups and Fields of Finite Morley Rank}
\begin{itemize}
\item  Definable Subgroups and Connected Components \cite[Ch. 5]{BN}
\item  Macintyre's Theorem \cite[Sec. 8.1]{BN}
\item  Zilber Indecomposability Theorem \cite[Ch. 5]{BN}
\item  Zilber's Definable Field Theorem \cite[Sec. 9.1]{BN}
\item  Finite Dimensionality \cite{La} % \nocite{Bo}
\end{itemize}


% \section{Deferentially Closed Fields}
% \begin{itemize}
% \item  Differential Ideals and Ritt-Raudenbush Basis Theorem \cite[II \S1]{MMP}
% \item  Differential Nullstellensatz and Quantifier Elimination \cite[II \S2]{MMP}
% \item  Elimination of Imaginaries \cite[II \S3]{MMP}
% \end{itemize}


\section{Algebraic Geometry}
\begin{itemize}
\item  Construction of Affine and Projective Schemes \cite[II \S2]{Ha}
\item  Construction of a Sheaf Associated to a Module \cite[II \S5]{Ha}
% \item  Basic Properties of Quasi-Coherent Sheaves \cite[II \S5]{Ha} % 5.6, 5.7
\item  Twisting Sheaf of Serre \cite[II \S5]{Ha}
\item  Vanishing of Cohomology for Flasque Sheaves \cite[III \S2]{Ha}
\item  Vanishing Theorem of Grothendieck \cite[III \S2]{Ha}
\item  Vanishing of Noetherian Affine Cohomology \cite[III \S2]{Ha}
\item  Computations with Cech Cohomology \cite[III \S4]{Ha}
\item  Equivalence of Cech and Normal Cohomology \cite[III \S4]{Ha}
\item  Cohomology of Projective Space \cite[III \S5]{Ha}
\end{itemize}


\bibliographystyle{alpha}

\def\cprime{$'$}
\begin{thebibliography}{MMP96}

\bibitem[BN94]{BN}
Alexandre Borovik and Ali Nesin.
\newblock {\em Groups of Finite {M}orley Rank}.
\newblock The Clarendon Press Oxford University Press, New York, 1994.
\newblock Oxford Science Publications.

\bibitem[CK73]{CK}
C.~C. Chang and H.~J. Keisler.
\newblock {\em Model theory}.
\newblock North-Holland Publishing Co., Amsterdam, 1973.
\newblock Studies in Logic and the Foundations of Mathematics, Vol. 73.

\bibitem[Har77]{Ha}
Robin Hartshorne.
\newblock {\em Algebraic geometry}.
\newblock Springer-Verlag, New York, 1977.
\newblock Graduate Texts in Mathematics, No. 52.

\bibitem[Hod97]{Ho}
Wilfrid Hodges.
\newblock {\em A Shorter Model Theory}.
\newblock Cambridge University Press, Cambridge, 1997.

\bibitem[Las98]{La}
Daniel Lascar.
\newblock Omega-stable groups.
\newblock In {\em Model theory and algebraic geometry}, pages 45--59. Springer,
  Berlin, 1998.

\bibitem[MMP96]{MMP}
D.~Marker, M.~Messmer, and A.~Pillay.
\newblock {\em Model theory of fields}.
\newblock Springer-Verlag, Berlin, 1996.

\bibitem[Pil83]{Pi1}
Anand Pillay.
\newblock {\em An Introduction to Stability Theory}.
\newblock The Clarendon Press Oxford University Press, New York, 1983.

\bibitem[Pil96]{Pi2}
Anand Pillay.
\newblock {\em Geometric Stability Theory}.
\newblock The Clarendon Press Oxford University Press, New York, 1996.
\newblock Oxford Science Publications.

\end{thebibliography}

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