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\title{Oral Qualifying Examination Syllabus}
\author{Kia Dalili}
\date{Committee: W. Vasconcelos, P. Landweber, F. Knop}

\maketitle



\section*{Algebraic Topology}

\begin{itemize}

\item[I.] Homology
\begin{enumerate}
\item Singular homology
\item Homotopy invariance
\item Mayer-Vietoris sequence
\item Relative homology groups
\item Excision 
\item Finite CW complexes
\item Skeletal homotopy
\item Eilenberg-Steenrod axioms
\end{enumerate}

\item[II.] Cohomology and products
\begin{enumerate}
\item Singular Cohomology
\item K\"unneth formula
\item Cup product
\item Hopf invariant
\item Cap product
\end{enumerate}

\item[III.] Manifolds and duality 
\begin{enumerate}
\item Manifolds and orientation
\item Thom isomorphism theorem
\item Poincar\'e duality
\end{enumerate}

\end{itemize}

\section*{Commutative Algebra}

\begin{itemize}

\item[I.] Basic commutative algebra
\begin{enumerate}
\item Localization of rings and modules 
\item The Hilbert Nullstellensatz
\item Associated primes and primary decomposition
\item Integral dependence and valuations
\item DVRs, Dedekind rings, and Krull rings
\item Graded rings and modules
\item Hilbert functions and Hilbert polynomials
\end{enumerate}

\item[II.] Cohen-Macaulay Rings
\begin{enumerate}
\item Regular sequences
\item Grade and depth
\item Depth and projective dimension
\item Cohen-Macaulay rings and modules
\item Complete intersections
\end{enumerate}

\item[III.] Homological Algebra
\begin{enumerate}
\item Chain complexes
\item Chain homotopies
\item Projective and injective resolutions
\item Left and right derived functors
\item Adjoint functors and left/right exactness
\item Tor and Ext
\item Dimensions
\item Koszul complexes
\item Local cohomology
\end{enumerate}

\end{itemize}



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