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\CEN{DARTMOUTH COLLEGE}
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\CEN{DEPARTMENT OF MATHEMATICS}
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\CEN{GRADUATE PROGRAM}
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\centerline{{\bf PROBABILITY:} {\em Syllabus for Graduate Certification\/}}
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The subject matter relevant to the graduate certification
requirement have been divided into primary and secondary topics.
The student is required to have a basic knowledge of the
three primary topics and several of the secondary topics,
the choice of which will be made together by examiners and examinee.

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\textbf{PRIMARY TOPICS: Required Basic Probability}
\begin{enumerate}

\item Basic discrete probability, including mean and variance, common distributions,
  Bayes' Theorem, Borel-Cantelli Lemma.

\item Basic continuous probability, including densities, joint distributions,
  measure theory, convolution, characteristic functions.

\item Central Limit Theorem, and some idea of its proof; weak and strong laws
  of large numbers.

\item Markov chains, discrete and continuous; stationarity; hitting and mixing times;
  recurrence and transience.

\item Information theory and entropy. 

\item Random walk, Brownian motion, law of the iterated logarithm.

\item Renewal processes, stopping times, renewal theorem.

\end{enumerate}

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\textbf{SECONDARY TOPICS: Additional Subjects in Probability}
\begin{enumerate}

\item Martingales and branching processes.

\item Statistical physics, Gibbs ensemble.

\item Randomized algorithms and their applications (e.g., primality testing).

\item The probabilistic method in combinatorics.

\item Probability amplitudes in quantum mechanics.

\item Random matrices and their eigenvalues.

\item Zero-one laws of Kolmogoroff and Fagin.

\item De Finetti's Theorem and finite versions.

\item Percolation.
  
\end{enumerate}

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\noindent REFERENCES
\begin{enumerate}
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\item Classical Combinatorics
  
  Bogart, {\em Introductory Combinatorics, Second Edition\/} (Chapters
  1,2,6)
  
  Graham, Rothschild, and Spencer, {\em Ramsey Theory\/} (Chapter 1)
  
  Liu, {\em Introduction to Combinatorial Mathematics\/}
  
  Riordan, {\em Combinatorial Mathematics\/} (Chapters 2---4)
  
  Stanley, {\em Enumerative Combinatorics, Volume 1\/} (Chapter 1)

\item  Algebraic Techniques
  
  Aigner, {\em Combinatorial Theory\/}
  
  Bogart, {\em Introductory Combinatorics, Second Edition\/} (Chapters
  3, 8)
  
  Stanley, {\em Enumerative Combinatorics, Volume 1\/} (Chapters
  1---3)
\item Graph Theory
  
  Bogart, {\em Introductory Combinatorics, Second Edition\/} (Chapters
  4,5)
  
  Bollobas, {\em Graph Theory: An Introductory Course\/}
  
  Bondy and Murty, {\em Graph Theory With Applications\/}
  
  Golumbic, {\em Algorithmic Graph Theory and Perfect Graphs\/}

\item[A.] Ordered Sets
  
  Birkhoff, {\em Lattice Theory\/}
  
  Davey and Priestley, {\em Introduction to Lattices and order\/}
  
  Bogart, {\em Introductory Combinatorics, Second Edition\/} (Chapter
  7)
  
  Stanley, {\em Enumerative Combinatorics, Volume 1\/}
  
  Trotter, {\em Combinatorics and Partially Ordered Sets: Dimension
    Theory\/} (Chapter 3)

\item[B.] Coding Theory
  
  Berlekamp, {\em Algebraic Coding Theory\/}
  
  Peterson and Weldon, {\em Error-Correcting codes\/}
  
  Pless, {\em Intorduction to the Theory of Error-Correcting Codes\/}
  
  Sloane and MacWilliams, {\em The Theory of Error Correcting Codes\/}
  
  Van Lint, {\em Coding Theory\/}

\item[C.] Combinatorial Geometry and Matroids
  
  Aigner, {\em Combinatorial Theory\/}
  
  Crapo and Rota, {\em On the Foundations of Combinatorial Theory:
    Combinatorial Geometries\/}
  
  Oxley, {\em Matroid Theory\/}
  
  Welsh, {\em Matroid Theory\/}

\item[D.] Matching Theory
  
  Bogart, {\em Introductory Combinatorics, Second Edition\/} (Chapter
  5)
  
  Hall, {\em Combinatorial Theory\/}
  
  Mirsky, {\em Transversal Theory: An Account of Some Aspects of
    Combinatorial Mathematics\/}
  
  Ryser, {\em Combinatorial Mathematics\/}

\item[E.] Random Graphs and the Probabilistic Method
  
  Bollobas, {\em Random Graphs\/}
  
  Palmer, {\em Graphical Evolution\/}
  
  Spencer, {\em Ten Lectures on the Probabilistic Method\/}

\item[F.] Symmetric Functions
  
  Macdonald, {\em Symmetric Functions and Hall Polynomials\/} (Chapter
  1)
  
  Sagan, {\em The Symmetric Group\/} (Chapters 3,4)
\item[G.] Representations of the Symmetric Group
  
  Sagan, {\em The Symmetric Group\/} (Chapters 1,2)
\end{enumerate}

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