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?)

• developer \neq user => lameness (e.g. gfan, polyhedral code)

• language, "relations", US-ification, Sage taking over the universe of free math software is dangerous, respect for other math software
• bugginess in core algebraic geometry code: not up to snuff, cannot trust the answers now, need to first fix the basics before implementing new stuff
• third party code currently not refereed: how do we know it's reliable?
• credit: young people, publication record
• longterm survivability
• keeping large scale vision, consistency among various development efforts/packages: categories, generic algorithms and info, learn from Axiom and MuPAD-combinat
• more documentation: to improve consistency, explain options to user (there can be 25 different algorithms for doing one thing)
• track usage

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

• all of the above
• very few developers (solution: M2 days, packages, journal)
• documentation and tutorial issues (improving)
• community building lacking
• uncertainty of long-term funding (only NSF, fickle)
• todo list needs to be upgraded (just text files in svn repository now)
• webpage needs to be upgraded
• narrow scope restricts number of developers (only commutative algebraists and algebraic geometers can contribute, and the intersection between this set and the set of software engineers/programmers is too small)
• narrow scope restricts users (e.g. lack of group theory functionality makes some algebraic geometers have to use a different system for their research)