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], and f^(1/n)
• GB's, free resolutions, flexible gradings, term orders
• rings (not necessarily commutative)
• modules (not just ideals, not just free)
• homological algebra
• linear algebra with basis an arbitrary index set I
• fast sparse and dense linear algebra over GF(q)

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)
• 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. GL_n-actions, functors, operations on functors, Yoneda product, tensor products)
• full homological algebra (spectral sequences, 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)
• super fast GB's and syzygies (speed and low memory usage)
• super fast primary 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 (lrs), double description
• optimization: linear and integer programming (coin/or), semidefinite programming
• combinatorial aspects
• polymake puts a lot of these things together, but it does not build!

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)