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Qualifying Exam Syllabus Sami H. Assaf Date: Monday, April 14, 2003

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, inclusion-exclusion;
exponential generating functions, Lagrange inversion, the Matrix-Tree theorem.
Posets and lattices: incidence algebra, M¨obius function and M¨obius inversion,
Eulerian posets, Euler characteristic, simplicial complexes, h-vectors, Cohen-
Macaulay complexes.
Symmetric functions: classical bases, Hall inner product, Schur functions, Robinson-
Schensted-Knuth correspondence, raising operators, Pieri formulas, Jacobi-Trudi
Reference: Stanley, Enumerative Combinatorics Vol. I & II
Major Topic: Commutative Algebra
Localization; Associated primes and primary decomposition; Integral depen-
dence, Nakayama's lemma, going-up, going-down; Hilbert's Nullstellensatz; Fil-
trations; Artin-Rees lemma; Completions, Hensel's lemma; Noether normaliza-
tion; Principle ideal theorem, systems of parameters; Valuation rings, discrete
valuation rings; Hilbert-Samuel polynomials; Cohen-Macaulay rings.
Reference: Eisenbud, Commutative Algebra with a View Toward Algebraic Geometry


Source: Assaf, Sami H. - Department of Mathematics, Massachusetts Institute of Technology (MIT


Collections: Mathematics