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\title{Qualifying Exam Syllabus}
\author{Josh Levenberg}
%\date{Sept 1, 1997}
\begin{document}
\maketitle

\section*{Algebraic Topology}
Math 215A
\bi
\li	Homotopy and the Fundamental group
\li	Covering spaces
	\bi\li	Homotopy/path lifting lemma
	\li	Properly discontinuous
	\li	Lens space $L(p,q)$
	\li	Monodromy / Deck translations
	\li	Semi-locally simply-connected
	\li	Universal cover
	\ei
\li	van Kampen's theorem
\li	Topological groups and H-spaces
\li	Simplical and CW complexes
	\bi\li	Cayley graph/complex
	\ei
\li	Homology
	\bi\li	Fundamental theorem of homological algebra
	\li	Simplicial
	\li	Singular
	\li	Cellular
	\li	Map from $\pi_1\to H_1$
	\li	Mayer-Vietoris sequence
	\li	Reduced
	\li	Relative
	\li	With coefficients
	\ei
\li	Cohomology
	\bi\li	Kronecker pairing
	\li	Cap and Cup product
	\li	Poincar\'e Duality
	\li	Orientation
	\li	Intersection theory (including transverseness)
	\li	Universal coefficient theorem
\ei\ei
Math 215B
\bi
\li	Homotopy in earnest
	\bi\li	Smash product
	\li	Loop space $\Omega$ and suspension $\Sigma$
	\li	Mapping cone
	\li	Whitehead product
	\ei
\li	Whitehead theorem
\li	Hurewicz theorem
\li	Eilenberg-MacLane space $K(\pi, n)$
\li	Fibrations
\li	Obstruction theory
\ei

\section*{Smooth Manifolds}
\textit{Differential Geometry Vol. I}, M. Spivak
\bi
\li	Ch 1. Manifolds (Definition, Surfaces, and Examples)
\li	Ch 2. Differentiable Structures
	\bi\li	$C^\infty$-related maps
	\li	Differentiable structure
	\li	Differentiable maps
	\li	Bump functions
	\li	Differentiation
	\li	Function rank
	\li	Critical/Regular Points/Values
	\li	Immersions and Embeddings
	\li	Submanifolds
	\li	The Shrinking Lemma
	\li	Partitions of unity
	\ei
\li	Ch 3. The Tangent Bundle
	\bi\li	Fibers
	\li	Tangent vectors
	\li	Vector bundles
	\li	Bundle map
	\li	Tangent bundle
	\li	Vector field
	\li	Orientation
		\bi\li	Orientation preserving or reversing maps
		\li	Orientation class
		\li	Orientable bundle
		\li	Orientable manifold
		\ei
	\ei	
\li	Ch 4. Tensors - only:
	\bi\li	Upper star
	\li	Dual bundle
	\ei
\li	Ch 5. Vector Fields and Differential Equations
	\bi\li	Integral Curves
	\li	Lipschitz condition
	\li	Contraction lemma
	\li	(Local) Flows
	\li	Lie bracket
	\ei
\li	Ch 6. Integral Manifolds
	\bi\li	Distributions (1-dimensional and k-dimensional)
	\li	Existance of integral manifolds
	\li	f-related vector fields
	\li	Integrable distributions
	\li	Frobenius Integrability Theorem
	\li	Foliations
	\ei
\ei
\textit{Morse Theory}, J. Milnor
\bi
\li	Transversality
\li	Stronger version of Sard's Theorem
\li	Finite dimensional Morse Theory
\ei

\section*{Combinatorics}
\textit{Enumerative Combinatorics Vol. I}, R. Stanley
\bi
\li	Counting
	\bi\li	Sets, multisets
	\li	Permutations
	\li	The Twelvefold Way
	\ei
\li	Inclusion-Exclusion
\li	Partially Ordered Sets
	\bi\li	Lattices
	\li	Distributive Lattices
	\li	Incidence Algebra
	\li	M\"obius Inversion
		\bi\li	Lattice M\"obius Algebras
		\li	M\"obius function Semimodular Lattices
		\ei
	\li	Zeta Polynomials
	\li	Rank-selection
	\li	Eulerian Posets
	\li	Binomial Posets and Generating Functions
	\ei
\li	Rational Generating Functions
	\bi\li	Theorem: Rational Power Series in One Variable
	\li	Cor: Polynomials
	\li	Linear Homogeneous Diophantine Equations
\ei\ei

\end{document}