Projects

Project ideas for [:days4:SAGE Days 4].

• W Stein: Documentation. Have teams describe how to do the following in tutorial fashion, with all examples automatically doctested:

  • Use dsage to task farm a couple of standard example problems.
  • Use SageX to create an interface to a C library (case study -- GMP-ECM).
  • Convert a program from .sage files to .py library code.
  • Convert a .py Python program to SageX for speed.
  • Create publication-quality graphics for inclusion in latex documents.
  • Port code from Magma to SAGE.
  • Make an existing PARI-GP / GAP / Singular / Lisp / Maxima program easily usable from SAGE.

• David Roe: General elliptic curve height bound code (port Cremona's GPL'd MAGMA code to SAGE.)

• W Stein: Implement Tate's algorithm over number fields (again, this involves porting Cremona's GPL'd MAGMA code to SAGE).

• W Stein, Tom Boothby, Dorian Ramier, Alex Clemesha, and Yi Qiang: Improve The SAGE Notebook:

  1. Convert the web server so that it uses twisted's multithreaded server (either twisted web like moinmoin or twisted web2)
  2. Make it so that the server use https to secure all communications and logins
  3. Make it so each user of the notebook has an account on the notebook, and can only see worksheets that are explicitly shared by other users. Worksheet names would internally be prefaced with the user name.
  4. Interface issues:
     1. Highlighting a block of text and pressing tab indents it four spaces; similarly, pressing shift-tab dedents it four spaces.
     2. Uploading a worksheet should not display an empty page.
  5. Create a worksheet settings panel; use this to:
     1. Set the math software system used to evaluate cells
     2. Determine whether pressing shift-enter jumps to the next cell or stays in current one.
     3. Determine whether or not the side bar is displayed (the left button wall could still be used to toggle it).
  6. Create a notebook settings panel:
     1. Whether or not panel is displayed by default
     2. Whether shift enter or enter evaluates a cell (Mathematica or Maple mode)
     3. Default CAS for new worksheets.
  7. Change "slide-show mode" to "single cell mode". Make it so the editor in this mode is a full-fledged code editor.
  8. Slide-show mode should flip through slides, which should be well-defined sections of the worksheet.
  9. Printing needs to be rewritten as follows: Given a single worksheet, print it by (1) saving it to a latex file with embedded images, and (2) running pdflatex on the result, then return the resulting pdf file and/or a tarball with the images and latex. This would be very high quality, and will also be something that would be easily included in latex documents. Printing a notebook (=all worksheets owned by a given user) would create a document with a section for each notebook.

  10. Rewrite saving/loading of worksheets, so it doesn't use pickle. Instead use the ''s text encoding and the images in directories.

  11. Finish implementing the html/edit mode: (1) images shouldn't get deleted on saving, (2) inserting new cells shouldn't be broken, and (3) it would be really nice, and probably easy, to have a WYSIWYG html editor mode like the one in moinmoin in which I'm writing this right now.
  12. Fix the bug where variables aren't listed in the sidebar.

• Nick Alexander: I'd like to implement some basic linear algebra over Z/nZ, following perhaps Algorithms for Linear Algebra Problems over Principal Ideal Rings (1996), Johannes Buchmann, Stefan Neis at http://citeseer.ist.psu.edu/719844.html. The algorithms of that paper apply to principal ideal rings, and in the case of a principal ideal domain, reduce to computing the Hermite normal form. So this could expand to trying to compute the Hermite normal form in the manner of Steel's unpublished algorithm as implemented in MAGMA.

• David Joyner: I'm interested in adding functionality to the group theory functionality in SAGE.

• David Joyner: I'm also interested in coding theory and in particular Robert Miller's work on (now GPL'd) Leon's partition backtracking programs.

• Kirsten Fagnan, Peter Clark: 3d Graphics

• Kirsten Fagnan, Josh Kantor: Numerical Methods

• David Roe: p-Adic Arithmetic -- Design and implement highly optimized algorithms for arithmetic with p-adic numbers. The current implementation is already extensive, containing a wide range of different models of p-adic arithmetic (capped relative, capped absolute, lazy, extensions, etc.) but many new algorithms need to be developed, especially for arithmetic with polynomials over the p-adics, for p-adic linear algebra, for computing in the lattice of extensions of a p-adic ring, and for computing with completions of a number field. (This project started at SAGE Days 2, and has been very actively pursued, especially by David Roe during the last 7 months.)

• Applications of Monsky-Washnitzer Cohomology: Investigate algorithms for fast point counting on elliptic and hyperelliptic curves, computation of p-adic height pairings, and computations of p-adic Coleman integrals. This project began in 2006 at an MSRI workshop, and has been an active area of work during the last year by Kiran Kedlaya, David Harvey, and Robert Bradshaw. In particular, Harvey has recently developed new algorithms that will be explored at the workshop, which allow for new efficient point counting on Jacobians of hyperelliptic curves, Bradshaw and Kedlaya have implemented Coleman integration which has application to the study of rational points on curves, and Stein has been investigating p-adic analogues of the Birch and Swinnerton-Dyer conjecture in many new cases using a new algorithm for computing p-adic heights.

• Quaternion Algebra Arithmetic and Modular Forms: Create and implement algorithms for efficient arithmetic in quaternion algebras, and apply this work to computation of Hecke operators, modular forms, Tamagawa numbers of modular abelian varieties, and enumeration of elliptic curves of large conductor. Gonzalo Tornaria has long worked in this area, and William Stein is working with David Kohel and Lassina Dembele on the design of better algorithms. This could also involve porting code from MAGMA to SAGE -- the two authors of Magma's quaternion algebra arithmetic have both strongly encouraged me to port their code to SAGE (or have partly done so themselves already).

• Distributed Parallel Computation: Continuing a major trend started at the MSRI workshop in January 2007 on parallel computation, participants will design algorithms for parallel distributed integer factorization, computation of Hecke eigenvalues, and other algorithms. In particular, participants will explore several of the other ideas listed above in the context of parallel computation. Implementing these algorithms will likely vastly improve the stability and reliability of distributed computation in SAGE.

• Optimized Polynomial Arithmetic: David Harvey and Bill Hart (Postdoc, Warwick) have recently created and implemented what is by far the world’s fastest code for univariate polynomial arithmetic. Explore how to fill in the remaining gaps in order to make the results of their work easily available to a larger user community. Similarly, Tom Boothby has worked for about 9 months on algorithms for very fast evaluation of polynomials at points, and another project will be to explore how to deploy this.

• Commutative Diagram Coercion Model: The current SAGE coercion model has some advantages (speed, ability to override for new classes) but also some issues. A user is not able to override the coercion system at runtime and use their own coercion maps (this could be incredibly useful for polynomials, or systems of field extensions). It is difficult to get a system-wide picture of what coercion maps exist: as SAGE grows, this will introduce bugs when the rules for transitivity of coercion are not followed. It is also difficult, though not impossible, to implement coercion systems where the computation of the coercion map requires effort (creating a coercion map from \mathbf{F}_{7^8} \to \mathbf{F}_{7^{24}}). I, David Roe, want to discuss a new coercion model, either replacing over built on top of the old, that addresses these problems. Currently I have some ideas for a system based on commutative diagrams, but I want to brainstorm some more before implementing it. This project will include discussions on ways to improve category theory in SAGE. Nick Alexander is also very interested in working on the proposed commutative diagram coercion model.