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== SAGE-related Problems == == Specific SAGE-related Problems ==
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1. [:msri07/threadsafety: Thread Safety of the SAGE Libraries]
2. [:msri07/pthread_sagex: Add Pthread support to SageX]
3. [:msri07/anlist: Implementation parallel computation of elliptic curve $a_p$ for SAGE]
 1. [[msri07/threadsafety| Thread Safety of the SAGE Libraries]]
 * [[msri07/pthread_sagex| Add Pthread support to SageX]]
 * [[msri07/anlist| Implementation in SAGE parallel computation of elliptic curve a_p for all p up to some bound]]
 * [[msri07/matrixadd| Implementation in SAGE matrix ADDITION over the rational numbers (say) using a multithreaded approach.]]
 * [[msri07/pointcount| Brute force count points on a variety over a finite field in parallel.]]
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== General Computer Algebra Problems == == Parallel Implementations ==
           
For each of the following, make remarks about how '''specific practical implementable''' parallel algorithms could be used to enhance mathematics software libraries (e.g., SAGE).

  *Arithmetic in Global Commutative Rings
     *The ring ${Z}$ of Integers
     *The ring ${Q}$ of Rational Numbers
     *Arbitrary Precision Real (and Complex) Numbers
     *Univariate Polynomial Rings
     *Number Fields
     *Multivariate Polynomial Rings
  *Arithmetic in Local Commutative Rings
     *Univariate Power series rings
     *$p$-adic numbers
  *Linear Algebra
     *Arithmetic of Vectors
          *Addition
          *Scalar Multiplication
          *Vector times Matrix
     *Rational reconstruction of a matrix
     *Echelon form
          *Echelon form over Finite Field
          *Echelon form over ${Q}$
          *Echelon form over Cyclotomic Fields
          *Echelon form (Hermite form) over ${Z}$
     *Kernel
          *Kernel over Finite Field
          *Kernel over ${Q}$
          *Kernel over ${Z}$
     *Matrix multiplication
          *Matrix multiplication over Finite Fields
          *Matrix multiplication over ${Z}$
          *Matrix multiplication over Extensions of ${Z}$
  *Noncommutative Rings
  *Group Theory
  *Groebner Basis Computation
  *Elliptic Curves
     *Generic elliptic curve operations
          *Group Law
          *Invariants
          *Division Polynomials
     *Elliptic curves over finite fields
          *Order of the group $E({{F}}_{p})$
          *Order of the group $E({{F}}_{q})$
          *Order of a point
     *Elliptic curves over ${{Q}}$ - part I
          *Birch and Swinnerton-Dyer Conjecture
          *Fourier coefficients
          *Canonical height of a point
          *Order of a point
          *Periods
          *Tate's algorithm
          *Conductor and Globally minimal model
          *CPS height bound
          *Torsion subgroup
          *Nagell-Lutz
          *An $l$-adic algorithm
          *Another $l$-adic algorithm
          *Mordell-Weil via 2-descent
          *Saturation
          *Heegner points
          *Heegner discriminants
          *Heegner Hypothesis
          *Heegner point index and height
     *Elliptic curves over ${{Q}}$ - part II
          *Root number
          *Special values of L-series
          *Sha bound
          *Isogenies
          *Attributes of primes
          *$p$-adic height
          *Modular Degree
          *Modular Parameterization
  *Hyperelliptic Curves
  *Modular Forms
     *Presentation of spaces of modular symbols
     *Hecke operators on modular symbols
     *Decomposition of spaces under the Hecke operators
     *Trace formulas
  *Computation of tables
     *Elliptic curves
     *Modular forms
     *Number fields
  *Cryptography
  *Coding Theory
  *Constants, functions and numerical computation

== John McKay CHALLENGE system of polynomial equations ==

http://www.cargo.wlu.ca/McKay/

MSRI 2007 Parallel Computation Problem List

  1. Thread Safety of the SAGE Libraries

  2. Add Pthread support to SageX

  3. Implementation in SAGE parallel computation of elliptic curve a_p for all p up to some bound

  4. Implementation in SAGE matrix ADDITION over the rational numbers (say) using a multithreaded approach.

  5. Brute force count points on a variety over a finite field in parallel.

Parallel Implementations

For each of the following, make remarks about how specific practical implementable parallel algorithms could be used to enhance mathematics software libraries (e.g., SAGE).

  • Arithmetic in Global Commutative Rings
    • The ring {Z} of Integers

    • The ring {Q} of Rational Numbers

    • Arbitrary Precision Real (and Complex) Numbers
    • Univariate Polynomial Rings
    • Number Fields
    • Multivariate Polynomial Rings
  • Arithmetic in Local Commutative Rings
    • Univariate Power series rings
    • p-adic numbers

  • Linear Algebra
    • Arithmetic of Vectors
      • Addition
      • Scalar Multiplication
      • Vector times Matrix
    • Rational reconstruction of a matrix
    • Echelon form
      • Echelon form over Finite Field
      • Echelon form over {Q}

      • Echelon form over Cyclotomic Fields
      • Echelon form (Hermite form) over {Z}

    • Kernel
      • Kernel over Finite Field
      • Kernel over {Q}

      • Kernel over {Z}

    • Matrix multiplication
      • Matrix multiplication over Finite Fields
      • Matrix multiplication over {Z}

      • Matrix multiplication over Extensions of {Z}

  • Noncommutative Rings
  • Group Theory
  • Groebner Basis Computation
  • Elliptic Curves
    • Generic elliptic curve operations
      • Group Law
      • Invariants
      • Division Polynomials
    • Elliptic curves over finite fields
      • Order of the group E({{F}}_{p})

      • Order of the group E({{F}}_{q})

      • Order of a point
    • Elliptic curves over {{Q}} - part I

      • Birch and Swinnerton-Dyer Conjecture
      • Fourier coefficients
      • Canonical height of a point
      • Order of a point
      • Periods
      • Tate's algorithm
      • Conductor and Globally minimal model
      • CPS height bound
      • Torsion subgroup
      • Nagell-Lutz
      • An l-adic algorithm

      • Another l-adic algorithm

      • Mordell-Weil via 2-descent
      • Saturation
      • Heegner points
      • Heegner discriminants
      • Heegner Hypothesis
      • Heegner point index and height
    • Elliptic curves over {{Q}} - part II

      • Root number
      • Special values of L-series
      • Sha bound
      • Isogenies
      • Attributes of primes
      • p-adic height

      • Modular Degree
      • Modular Parameterization
  • Hyperelliptic Curves
  • Modular Forms
    • Presentation of spaces of modular symbols
    • Hecke operators on modular symbols
    • Decomposition of spaces under the Hecke operators
    • Trace formulas
  • Computation of tables
    • Elliptic curves
    • Modular forms
    • Number fields
  • Cryptography
  • Coding Theory
  • Constants, functions and numerical computation

John McKay CHALLENGE system of polynomial equations

http://www.cargo.wlu.ca/McKay/

msri07/problems (last edited 2008-11-14 13:42:04 by anonymous)