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 * fast R[x_1,...,x_n], and f^(1/n)
 * GB's, free resolutions, flexible gradings, term orders
 * fast R[x_1,...,x_n], also with fractional exponents
 * GB's, flexible gradings, term orders
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 * homological algebra  * sheaves
* homological algebra (free resolutions)
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 * fast sparse and dense linear algebra over GF(q)  * fast sparse and dense linear algebra over finite fields
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 * (huge) polyhedral geometry (not just polymake)  * (huge) polyhedral geometry (not just polymake, see more below)
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 * full functoriality (e.g. GL_n-actions, functors, operations on functors, Yoneda product, tensor products)
 * full homological algebra (spectral sequences, etc.)
 * full functoriality (e.g. preservation of GL_n-actions, functors, operations on functors, Yoneda product, tensor products)
 * full homological algebra (spectral sequences, derived categories, etc.)
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 * a "good clean" programming language (not just M2, e.g. Maple -- having to put things into rings before being able to use them is annoying)  * a "good clean" programming language (not just M2, e.g. Maple -- having to put things into rings before being able to use them is annoying; work easily with general expressions)
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 * super fast primary decomposition (e.g. numerical) and integral closure  * super fast and low memoryprimary decomposition (e.g. numerical) and integral closure
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 * optimal performance: important algorithms are reverse search (lrs), double description
 * optimization: linear and integer programming (coin/or), semidefinite programming
 * optimal performance: important algorithms are reverse search (as in lrs, uses less memory), double description (track the duals, as in cdd and 4ti2)
 * optimization: linear and integer programming (coin/or), semidefinite programming (any good software for this?)

Discussion: Sage, Macaulay 2, and other Mathematical Software for Algebraic Geometry

What are the absolutely critical features that you '''must''' have in the mathematical software you use for '''your''' research? (E.g., fast linear algebra, Groebner basis, sheaves?)

  • modular forms
  • fast R[x_1,...,x_n], also with fractional exponents
  • GB's, flexible gradings, term orders
  • rings (not necessarily commutative)
  • modules (not just ideals, not just free)
  • sheaves
  • homological algebra (free resolutions)
  • linear algebra with basis an arbitrary index set I
  • fast sparse and dense linear algebra over finite fields

What are the '''killer features''' that your dream mathematical software would have? (e.g., good mailing list, free, super fast algorithm for XXX, latex output?)

  • (huge) polyhedral geometry (not just polymake, see more below)
  • representation theory for finite groups (char 0 and modular, not just GAP, compare to what MAGMA can do -- and how fast it can do it)
  • rings of representations (Grothendieck rings, etc.)
  • local rings and global rings: localization, really working (not just M2)
  • GB over all rings (e.g. field extensions), even noncommutative when possible
  • full functoriality (e.g. preservation of GL_n-actions, functors, operations on functors, Yoneda product, tensor products)
  • full homological algebra (spectral sequences, derived categories, etc.)
  • parallelize everything
  • deformation theory
  • a "good clean" programming language (not just M2, e.g. Maple -- having to put things into rings before being able to use them is annoying; work easily with general expressions)
  • super fast GB's and syzygies (speed and low memory usage)
  • super fast and low memoryprimary decomposition (e.g. numerical) and integral closure
  • sheaves, Chern classes, intersection theory on singular spaces
  • algebraic topology on complex and real points on a variety
  • etale cohomology
  • usable resolution of singularities

Polytopes

  • packages: lrs, cddlib, porta, 4ti2, polymake, coin/or
  • optimal performance: important algorithms are reverse search (as in lrs, uses less memory), double description (track the duals, as in cdd and 4ti2)
  • optimization: linear and integer programming (coin/or), semidefinite programming (any good software for this?)
  • combinatorial aspects
  • polymake puts a lot of these things together, but it does not build!

Linear algebra

  • for a lot of the above issues, we need very very fast linear algebra over a huge range of fields/rings

What are some things that disturb you about the direction in which Sage is going? (E.g., too big/ambitious? not open enough or too open? too many bugs? changing too quickly? referee process for code inclusion too onerous?)

What are some things that disturb you about the direction in which Macaulay2 is going? (similar e.g. as above)

days14/what (last edited 2009-03-11 20:47:45 by AlexGhitza)