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Coding sprint and project ideas for [:days4:SAGE Days 4] June 1217, 2007.  
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Project ideas for [:days4: SAGE Days 4].  Each section lists the project followed by the main contact person for that project. 
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* 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. I'm also very interested in working on the commutative diagram coercion model proposed by David Roe.  * [:days4/projects/packaging: Packaging and Distributing SAGE]  Didier Deshommes * [:days4/projects/nb: The SAGE Notebook]  William Stein 
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* David Joyner: I'm interested in adding functionality to the group theory functionality in SAGE. I'm also interested in coding theory and in particular Robert Miller's work on (now GPL'd) Leon's partition backtracking programs.  * [:days4/projects/dist: Distributed Computation]  Yi Qiang 
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* pAdic Arithmetic: Design and implement highly optimized algorithms for arithmetic with padic numbers. The current implementation is already extensive, containing a wide range of different models of padic arithmetic (capped relative, capped absolute, lazy, extensions, etc.) but many new algorithms need to be developed, especially for arithmetic with polynomials over the padics, for padic linear algebra, for computing in the lattice of extensions of a padic 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.)  * [:days4/projects/doc: Documentation]  ??? 
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* Applications of MonskyWashnitzer Cohomology: Investigate algorithms for fast point counting on elliptic and hyperelliptic curves, computation of padic height pairings, and computations of padic 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 padic analogues of the Birch and SwinnertonDyer conjecture in many new cases using a new algorithm for computing padic heights.  * [:days4/projects/numerical: Numerical Computation]  Josh Kantor 
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* 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.  * [:days4/projects/arith: Basic Arithmetic ]  David Harvey 
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* 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.  * [:days4/projects/numbertheory: Number Theory]  David Roe 
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* 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.  * [:days4/projects/coding: Coding Theory]  Robert Miller 
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* 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 systemwide 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 GF(7^8) > GF(7^24)). I 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.  * [:days4/projects/groups: Group Theory]  David Joyner * [:days4/projects/linalg: Linear Algebra]  Nick Alexander * [:days4/projects/combinat: Combinatorics]  Mike Hansen (remote) * [:days4/projects/calculus: Symbolic Calculus]  Bobby Moretti * [:days4/projects/graphs: Graph Theory]  Robert Miller * [:days4/projects/3d: 3d Graphics]  Robert Bradshaw 
Projects
Coding sprint and project ideas for [:days4:SAGE Days 4] June 1217, 2007.
Each section lists the project followed by the main contact person for that project.
 [:days4/projects/packaging: Packaging and Distributing SAGE]  Didier Deshommes
 [:days4/projects/nb: The SAGE Notebook]  William Stein
 [:days4/projects/dist: Distributed Computation]  Yi Qiang
 [:days4/projects/doc: Documentation]  ???
 [:days4/projects/numerical: Numerical Computation]  Josh Kantor
 [:days4/projects/arith: Basic Arithmetic ]  David Harvey
 [:days4/projects/numbertheory: Number Theory]  David Roe
 [:days4/projects/coding: Coding Theory]  Robert Miller
 [:days4/projects/groups: Group Theory]  David Joyner
 [:days4/projects/linalg: Linear Algebra]  Nick Alexander
 [:days4/projects/combinat: Combinatorics]  Mike Hansen (remote)
 [:days4/projects/calculus: Symbolic Calculus]  Bobby Moretti
 [:days4/projects/graphs: Graph Theory]  Robert Miller
 [:days4/projects/3d: 3d Graphics]  Robert Bradshaw