Computational Commutative Algebra (and Algebraic Geometry)

    MATH 531 -- Fall 2004 (Call #20093)

    WHEN = 10am
    WHERE = SEO 512

  • Description:

    • Advances in computing over the last couple of decades have revolutionized the area of commutative algebra and algebraic geometry, making tractable many problems inaccessible in the past, and providing powerful tools for experimentation.

    This course serves a dual purpose: its objective is to introduce the basic concepts of commutative algebra, as well as provide hands-on experience with computer algebra software. Besides regular lectures, there would be (either weekly or biweekly) lab sessions, where we would use Maple and Macaulay 2 to solve computational problems.

  • Prerequisites:

    • The course is self contained; firm knowledge of linear algebra is the only requirement. The material should be accessible for both pure math and MCS graduate students.

  • Texts:
    • (main) Cox, Little, O'Shea, ``Using algebraic geometry'', Springer GTM, 1998.
    • Cox, Little, O'Shea, ``Ideals, varieties, and algorithms'', Springer UTM, 1997.
    • Eisenbud, ``Commutative Algebra with a view toward algebraic geometry'', Springer GTM, 1995.
    • Edited by Eisenbud, Grayson, Stillman and Sturmfels, ``Computations in algebraic geometry with Macaulay 2'', Springer, ``Algorithms and Computations in Mathematics'' #8, 2001
    • Greuel, Pfister, ``A SINGULAR Introduction to Commutative Algebra'', Springer, 2002.

     

  • Links:
    • Macaulay 2
    • MAPLE
    • commalg.org -- Commutative Algebra portal
    • Basic Algebraic Geometry -- collection of lecture notes

     

  • Tentative topics:
    • polynomials, ideals, varieties
      • monomial orders
      • Gröbner bases
      • affine varieties
      • algebra-geometry dictionary
      • primary decomposition
      • polynomial and rational functions on a variety
    • solving systems of polynomial equations
      • solving via elimination
      • Gröbner basis conversion (FGLM)
      • solving via eigenvalues
      • finding real roots
    • resultants
      • multivariate resultants
      • solving via resultants
    • local rings
      • multiplicities and Milnor numbers
      • term orders
      • standard bases
    • modules
      • monomial orders and Gröbner bases
      • syzygies
      • modules over local rings
    • polytopes
      • geometry of polytopes
      • Minkowski Sums and mixed volumes
      • Bernstein's theorem