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Oral Syllabus of David Nacin
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Minor Topic:  The Automorphism Tower Problem

Simon Thomas: The Automorphism Tower Problem
\begin{enumerate}
\item The Automorphism Tower Problem
\begin{enumerate}
\item The Automorphism Tower Problem
\item Some Fundamental Results
\item Some Examples of Automorphism Towers
\item The Infinite Dihedral Group
\end{enumerate}
\item Wielandt's Theorem
\begin{enumerate}
\item Automorphism Towers of Finite Groups
\item Subnormal Subgroups
\item Finite p-gorups
\end{enumerate}
\item Some Examples of Automorphism Towers
\begin{enumerate}
\item The Automorphism Tower Theorem
\item $T_k$ is increasing
\item A Better Bound
\item The Automorphism Tower Problem Revisited
\end{enumerate}
\item The Normalizer Tower Technique
\begin{enumerate}
\item Normalizer Towers
\item Coding Structures in Graphs
\item Coding Graphs in Fields
\item A Technical Lemma
\end{enumerate}
\item Hamkins' Theorem
\begin{enumerate}
\item Automorphism Towers of Arbitrary Groups
\item Some Examples
\end{enumerate}
\end{enumerate}
Major Topic:  Lie Algebras and Representations

James Humphreys:  Introduction to Lie Algebras and Representation Theory
\begin{verse}
\item 1. Definitions and First Examples
Definitions and Classical Algebras,
Lie Algebras of Derivations,
Structure Constants and Abstract Lie Algebras
\item 2. Ideals and Homomorphisms,
Important Ideals,
Homorphisms and Representations,
Automorphisms
\item 3. Solvable and Nilpotent Lie Algebras,
Solvability and Nilpotency,
Engel's Theorem
\item 4. Theorems of Lie and Cartan,
Lie's Theorem,
Jordan-Chevalley Decomposition,
Cartan's Criterion
\item 5. Killing Form,
Criterion for Semisimplicity,
Inner Derivations,
Abstract Jordan Decomposition
\item 6. Complete Reducibility of Representations,
Modules,
Casmir Element of a Representation,
Weyl's Theorem,
Preservation of Jordan Decomposition
\item 7. Representations of sl(2,F),
                Weights and Maximal Vectors,
                Classification of Irreducible Modules
\item 8. Root Space Decomposition,
Maximal Toral Subalgebras and Roots,
Centralizer of H,
Orthogonality Properties,
Integrality Properties,
Rationality Properties

\item 9. Axiomatics, Reflections in a euclidean space, root systems,
examples, pairs of roots

\item 10. Simple Roots and Weyl group, Bases and Weyl chambers,
Lemmas on simple roots, The Weyl group, Irreducible root systems

\item 11. Classification, Cartan matrix of $\Phi$, Coxeter graphs and
Dynkin diagrams, Irreducible components, Classification theorem

\item 12. Construction of root systems and automorphisms, Construction
of types A-G, Automorphisms of $\Phi$

\item 13. Abstract theory of weights, Weights, Dominant Weights, The
Weight $\delta$, Saturated sets of weights

\item 18. Generators and Relations,
Relations Satisfied by L and Consequences,
Serre's Theorem,
Application: Existence and Uniqueness
\item  20. Weights and Maximal Vectors, 
Weight Spaces, Standard Cyclic Modules,
Existence and Uniqueness
\item  21. Finite Dimensional Modules, Necessary Condition, Sufficient
Condition, Weight Strings and Weight Diagrams, Generators and Relations for $V(\lambda)$
\item 22. Multiplicity Formula, A Universal Casmir Element, Traces on Weight Spaces,
     Freudenthal's Formula, Examples, Formal Characters
\item 23. Characters,
     Invariant Polynomial Functions,
     Standard Cyclic Modules and Characters,
     Harish-Chandra's Theorem
\item 24. Formulas of Weyl, Kostant, and Steinberg,
     Some Functions on $H^*$,
     Kostant's Multiplicity Formula,
     Weyl's Formulas,
     Steinberg's Formula
\end{verse}

James Lepowsky:  Lectures on Kac-Moody Lie Algebras
\begin{enumerate}
\item   Kac-Moody Lie Algebras
\item   The Weyl Group
\item   Standard Modules
\item   The Numerator Formula
\item   The Symmetrizable Case and the Invariant Form
\item   The Casimir Operator and Consequences
\item   The Weyl-Macdonald-Kac Formulas
\end{enumerate}


Nathan Jacobson:  Lie Algebras
\begin{itemize}
\item Complete Reducibility of the Representations of Semisimple Algebras
\item Cohomology Groups of a Lie Algebra
\item Definition and Basic Properties of Universal Enveloping Algebras
\item The Poincare-Birkhoff-Witt Theorem
\item Filtration and Graded Algebras
\item Free Lie Algebras
\item Cohomology of Lie Algebras
\end{itemize}

\begin{verse}
Igor Franknel/James Lepowsky/Arne Meurman:  Vertex Operator Algebras and
the Monster
\item 1.  	Lie Algebras\\
  1.1     Algebras\\
  1.2     Modules\\
  1.3     Algebra Constructions\\
  1.4     Module Constructions\\
  1.5     Induced Modules\\
  1.6     Affine Lie Algebras\\
  1.7     Heisenberg Algebras\\
  1.8     Contravariant Forms\\
  1.9     The Virasoro Algebra\\
  1.10    Graded Dimension\\
\item 2.      Formal Calculus\\
  2.1     Formal Series\\
  2.2     Derivations\\
  2.3     Affine Lie Algebras Via Formal Variables
\item 3.      Realizations of $sl(2)^{\wedge}$ by Twisted Vertex Operators\\
  3.1     The Affine Lie Algebra $sl(2)^{\wedge}$\\
  3.2     The Twisted Vertex Operators $X_{Z+1/2}(\alpha, z)$\\
  3.3     Normal Ordering\\
  3.4     Some Commutators
\item 4.      Realizations of $sl(2)^{\wedge}$ by Untwisted Vertex Operators\\
  4.1     2-Cocycles\\
  4.2     Normal Ordering\\
  4.3     Some Commutators\\
  4.4     Irreducible Representations of $sl(2)^{\wedge}$ and $sl(2)^{\wedge}[\theta_1]$
\item 5.      Central Extensions\\
  5.1     2-Cocycles\\
  5.2     Commutator Maps\\
  5.3     Extraspecial 2-Groups\\
  5.4     Automorphisms of Central Extensions\\
  5.5     Representations of Central Extensions
\item 6.      The Simple Lie Algebras $A_n$, $D_n$, $E_n$\\
  6.1     Lattices\\
  6.2     A Class of Lie Algebras\\
  6.3     The Cases $A_n$, $D_n$, $E_n$\\
  6.4     A Group of Automorphisms of g
\end{verse}
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