= MSRI 2007 Parallel Computation Problem List =

== Specific SAGE-related Problems ==

 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.]]

== 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/