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\begin{document}
James Taylor's Oral Exam Syllabus
\begin{tabbing}

I. \= Functional Analysis \\

\> A. \= Basics of Banach spaces \\
\> \> i. Examples such as $L^p$ spaces, sequences spaces, direct sums,
and quotients \\
\> \> ii. Linear functionals: duals, reflexive spaces, and the Hahn
Banach theorem \\
\> \> iii. Baire category, open maps, closed graphs, and Banach-Steinhaus \\
\> \> iv. Basics of Hilbert spaces (polarization, adjoints, Riesz
lemma) \\

\> B. Useful topological notions \\ 
\> \> i. Nets \\ 
\> \> ii. Compactness (Tychonoff's \Thm, Urysohn's Lemma,
Stone-Weierstrass \Thm) \\
\> \> iii. Measure theory on compact spaces (Riesz-Markov \Thm) \\
\> \> iv. Various topologies on operator spaces \\
\> \> v. The Banach-Alaoglu \Thm \\

\> C. Bounded Operator Theory \\
\> \> i. Adjoints \\
\> \> ii. Spectrum \\
\> \> iii. Positive operators, square roots \\ 
\> \> iv. Compact operators \\
\> \> v. Fredholm operators and the Fredholm Alternative \\
\> \> vi. Spectral Mapping Theorem \\
\> \> vii. Functional Calculus \\
\> \> viii. Various Spectral Theorems and measures related to
self-adjoint operators\\

\> D. Unbounded Operator Theory  \\
\> \> i. Definitions and generalizations \\
\> \> ii. Self-adjoint, closed, essentially
self-adjoint, and symmetric operators\\
\> \> iii. Cayley Transform \\
\> \> iv. Spectral Theorem for unbounded self-adjoint operators\\
\> \> v. Stone's Theorem \\

\> E. Differential Calculus on Banach Spaces \\
\> \> i. Derivative of operators on Banach spaces and generalizations
of elementary differential calculus \\
\> \> ii. Inverse and Implicit Function \Thms \\
\> \> iii. Infinite Dimensional Manifolds and their issues \\

\> F. Differential Operators and Spectral Theory \\
\> \> i. Schwarz space \\
\> \> ii. Fourier Transform \\
\> \> iii. Distributions \\
\> \> iv. Sobolev Spaces \\
\> \> v. Various Laplace operators and their uses \\

\> G. Applications to Quantum Mechanics \\
\> \> i. Basics of Bohmian Mechanics and the formalism of Quantum
Mechanics \\
\> \> ii. Position, momentum operators \\
\> \> iii. One-dimensional problems such as the harmonic osciallator,
different potential wells \\
\> \> iv. Spin\\
\> \> v. Bell's Theorem \\

II. \=Differential Geometry \\

\> A. \= Basic Definitions and examples \\
\> \> i. Definitions of Manifolds, tangent vectors, vector fields,
vector bundles\\
\> \> ii. Examples: Surfaces, Lie groups-Matrix groups, submanifolds\\
\> \> iii. Various mappings such as immersions, induced maps \\
\> \> iv. Quotient manifolds: Projective spaces, Grassmann manifolds\\

\> B. Tensors and differential forms \\
\> \> i. Tensors of all types, tensor fields, maps and tensors \\
\> \> ii. Exterior algebra, exterior derivative, differential forms \\
\> \> iii. Orientability and n-forms \\
\> \> iv. Symmetrizing, alternating, contracting, and multiplying tensors\\
\> \> v. Tensor Derivations \\
\> \> vi. Lie Derivatives \\
\> \> vii. Poincare Lemma and its partial converse \\

\> C. Vector fields\\
\> \> i. Existence and Uniqueness \Thms\ for ODE \\
\> \> ii. One-Parameter groups \\
\> \> iii. Vector Fields as flows and as differential operators \\
\> \> iii. Lie algebra of vector fields \\
\> \> iv. Frobenius' \Thm\ and foliations \\

\> D. Metrics and Connections \\
\> \> i. Definition of Metrics and Connections \\ 
\> \> ii. Covariant Derivative \\
\> \> iii. The Levi-Civita connection \\
\> \> iv. Parallel Translation \\
\> \> v. Geodesics \\
\> \> vi. Frame fields \\
\> \> vii. Exponential Map \\
\> \> viii. Hopf-Rinow \Thm \\
\> \> ix. Curvature \\

\> E. Integration on Manifolds \\
\> \> i. Definition of the integral \\
\> \> ii. Manifolds with boundary \\
\> \> iii. Stokes' \Thm \\

\> F. Surface theory \\
\> \> i. Fundamental Forms, Gauss Curvature, Principal Curvature \\
\> \> ii. The Gauss Theorem \\
\> \> iii. Gauss-Bonnet Theorem \\ 
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\end{document}







