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\centerline{\textbf{ Oral Qualifying Examination Syllabus}}
\centerline{\textbf Sarah Genoway }
\centerline{April 28, 2005}
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\centerline{Committee: D. Maclagan, W. Vasconcelos, C. Weibel, D. Zeilberger}
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\section{Commutative Algebra}
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\subsection{Basic Constructions}
\begin{itemize}
	\item {Localization}
	\item {Hom, Tensor, Ext, and Tor}
  \item {Associated Primes and Primary Decomposition of Modules}
  \item {Integral dependance and Integral Extensions}
  \item {Graded Rings and Modules}
  \item {Filtrations}
  \begin{itemize}
  \item{I-stable filtration}
  \item{Associated Graded Ring and Module}
  \item{Blow-up Algebra, Rees Algebra}
  \item{Artin-Rees Lemma}
  \item{Krull Intersection Theorem}
  \end{itemize}
  \item {Completions and Hensel's lemma}
  \end{itemize}
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\subsection{Dimension Theory}

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\begin{itemize}
	\item {Dimension Zero}
	\item {Principal Ideal Theorem and System of Parameters}
	\item {Noether Normalization}
	\item {Finiteness of Integral Closure}
\end{itemize}

\subsection{Cohen Macaulay Rings}
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\begin{itemize}
	\item {Regular Sequences}
	\item {Grade, Depth, and Projective Dimension}
	\begin{itemize}
	\item {Auslander-Buchsbaum Formula}
	\end{itemize}
	\item {Cohen-Macaulay Rings and Modules}
	\item {Regular Local Rings}
	\item {Normal Rings}
	\begin{itemize}
	\item{R1 and S2 conditions}
	\end{itemize}
	\item {Complete Intersections}
	\end{itemize}

\pagebreak
\section{Computational Geometry}
\subsection{Monomial Ideals}
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\begin{itemize}
	\item {Dickson's Lemma}
	\item {Primary Decompostition}
	\item {Cellular Resolution}
	\item {Multi-gradings}
	\item {Borel-Fixed Ideals}
\end{itemize}

\subsection{Gr\"{o}bner Bases}

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\begin{itemize}
	\item {Monomial Orders}
	\item {Weight Orders}
	\item {Buchberger's Criterion}
	\item {Buchberger's Algorithm}
	\item {Free Resolutions}
	\begin {itemize}
	\item {Schreyer's Method for Computing Syzygies}
	\item {Hilbert Syzygy Theorem}
	\item {Minimal Free Resolutions}
	\end{itemize}
	\item {Universal Gr\"{o}bner Basis}
	\item {State Polytope and Grobner Fan}
	
\end{itemize}

\subsection{Hilbert Functions}
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\begin{itemize}
	\item {Bezout's Theorem}
	\item {Macaulay's Theorem}
	\item {Gotzmann's Regularity and Persistence Theorems}
\end{itemize}

\subsection{Varieties}

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\begin{itemize}
	\item {Nullstellensatz}
	\item {Ideal-Variety Dictionary}
	\item {Elimination and Projection}
	\item {Toric Ideals and Varieties}
	\begin{itemize}
	\item{Toric Ideal is Generated by Binomials}
	\item{Compute Hilbert Basis for Affine Toric Variety}
\end{itemize}
\end{itemize}
\subsection{Polytopes}

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\begin{itemize}
	\item {Equivalent Characterizations}
	\item {Simple and Simplicial Polytopes}
	\item {Cyclic Polytope and Gale's Evenness Condition}
	\item {Simplicial Complex}
	\item {Shellability}
	\item {Upper Bound Theorem for Convex Polytopes}
\end{itemize}

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%\begin{thebibliography}{99}
%	\bibitem{AM}{Atiyah, M.F. and Macdonald, I.G., (1969).
%  \textit{Introduction to Commutative Algebra}. Addison-Wesley.}
%	\bibitem{BH}{Bruns, W. and Herzog, J. (1993). \textit{Cohen-Macaulay Rings}. Cambridge University Press.}
%	\bibitem{E}{Eisenbud, D. (1994). \textit{Commutative Algebra with a
%	View Toward Algebraic Geometry}. Springer-Verlag.}
%	\bibitem{WF}{Fulton, W. (1993). \textit{Introductions to Toric Varieties}. Princeton University Press.  	Chapter 1.}
%	\bibitem{BS}{Sturmfels, B. (1996).
%	\textit{Grobner Bases and Convex Polytopes}, American Mathematical Society, Univ. Lectures Series, No 8.}
%	\bibitem{GZ}{Zeigler, G. (1995). \textit{Lectures on Polytopes}.  Springer-Verlag.}
	
%\end{thebibliography}
\end{document}
