 
Summary: Qualifying Exam Syllabus Sami H. Assaf
Date: Monday, April 14, 2003
Committee: D. Eisenbud, M. Haiman, D. Sarason (chair), T. Speed (Statistics)
Major Topic: Combinatorics
Counting: ordinary generating functions, the Twelvefold Way, inclusionexclusion;
exponential generating functions, Lagrange inversion, the MatrixTree theorem.
Posets and lattices: incidence algebra, M¨obius function and M¨obius inversion,
Eulerian posets, Euler characteristic, simplicial complexes, hvectors, Cohen
Macaulay complexes.
Symmetric functions: classical bases, Hall inner product, Schur functions, Robinson
SchenstedKnuth correspondence, raising operators, Pieri formulas, JacobiTrudi
identity.
Reference: Stanley, Enumerative Combinatorics Vol. I & II
Major Topic: Commutative Algebra
Localization; Associated primes and primary decomposition; Integral depen
dence, Nakayama's lemma, goingup, goingdown; Hilbert's Nullstellensatz; Fil
trations; ArtinRees lemma; Completions, Hensel's lemma; Noether normaliza
tion; Principle ideal theorem, systems of parameters; Valuation rings, discrete
valuation rings; HilbertSamuel polynomials; CohenMacaulay rings.
Reference: Eisenbud, Commutative Algebra with a View Toward Algebraic Geometry
