Sage Days 15 Project Ideas

These are project ideas for Sage Days 15

MSVC 2008 port of PARI-2.4.2.alpha0

• Do a port of PARI-2.4.2.alpha0 to MSVC 2008, including MPIR support.

Low Level Simplicial Sets, CW Complexes, and Morphisms for Simplicial Complexes and Chain Complexes

PEOPLE: S. Ault, B. Antieau

• Rudimentary method to check if input defines a simplicial set. Brainstorming on how to make this accessible to a user.

• S. Ault has a CW Complex class up and running (including the chain_complex() method). The next step is to work on a homology_with_generators method within the ChainComplex (See trac #5882 and #6100).

  • Constructor of CW complexes from simplicial complexes.
  • Will include some standard CW complexes for playing around with. Including various projective spaces.
• B. Antieau has simplicial complex morphisms, and has the functor to morphisms of chain complexes. (See trac #6099.)
  • There are simplicial complex morphism and chain complex morphism classes, and there is a method on the former to produce and instance of the latter, including base rings, augmentation, chain v. cochain.
  • Working on various associated complexes of a map, like graph, mapping_{cone,cylinder,path_space}.
  • Working on fiber products.
  • Prepping for induced maps on homology, cohomology, ring structure on cohomology. (See trac #6101 and #6102.)
  • Milestone for these patches is 4.0.2.
• All of this code is going rely heavily on William's #5882 patch for arbitrary finitely generated modules over rings with implemented Smith normal form. As the SAGE code for linear algebra gets faster, we will be able to compute more and more homology and cohomology.
• This all provides groundwork for the creation of modules over John Palmieri's SAGE implementation of the Steenrod algebra, which is a long-term goal.

Prime Pi

PEOPLE: Fredrik Johansson, William Stein, Robert Bradshaw

• Investigate implementing the Meissel, Lehmer, Lagarias, Miller, Odlyzko for computing the number of primes up to x.

Steenrod Algebra cohomology (Yacop integration)

PEOPLE: Christian, John Palmieri, S. Ault, Bob Bruner

• A first version is already available. Here's a page with installation instructions: Yacop

Stuff for the final report:

• Installation problems on Mac OS X fixed
• Defined SAGE framework for modules over the Steenrod algebra and created a first proof-of-concept implementation of some modules.

MinRes - Mininmal resolutions over the Steenrod algebra

PEOPLE: Bob Bruner, Christian Nassau, John Palmieri, David Roe

• Create SAGE interface to Bruner's minimal resolutions package
• Package is general for mod 2 connected algebras, but currently only has

  • support for the mod 2 Steenrod algebra and the subalgebra <Sq^1, Sq^2, Sq^4>

• Focus is on multiplicative structure: Ext_A(M,M) as an algebra w.r.t. Yoneda

  • product, and Ext_A(M,N) and Ext_A(N,M) as modules over it.

Accomplished this week:

• Crude Steenrod algebra as vector space datatype and interface to module creation and
  • minimal resolution calculation
• Results integrated into Yacop's GUI and database manager.

Still to do:

• Integrate crude datatype into better Steenrod module datatype

• Implement MinRes's FP module datatype

• Implement chain maps (MinRes can lift an arbitrary cocycle - from this the multiplicative

  • structure is evident, including Toda brackets)
• Create spectral sequence datatype to accept Ext algebras and modules as inputs, and

  • differentials (calculated using Bruner's thesis in case of H_\infty ring spectra) (lazy evaluation essential here) and use algebra structure to extend and explore consequences.

• Use Palmieri's Steenrod algebra routines to minimize effort in describing modules

Quaternion Algebras (implement odd power of p dividing level case)

PEOPLE: Alia Hamieh, Jerome Lefebvre, William Stein.

• working on it.

Sage/Magma Comparison Audit

PEOPLE: William Stein, Bill Hart, Robert Bradshaw, Alia Hamieh, Jerome Lefebvre.

• See sagebeatsmagma and magma

• Alia compared all of the functionality in both quadratic forms package and algebraic geometry package. She also has implemented an algorithm to be included in the binary quadratic form package.
• Jerome has done so for representation theory and is working on improving the group algebra in sage.

And we are both done with comparing. And we have both submitted the changes we wanted to make on this.

CANCELLED (already in Sage!) Generalized Bernoulli Numbers

PEOPLE: William Stein

• There is a simple 2 or 3 line formula for generalized Bernoulli numbers in terms of non-generalized ones, which could be used to speed up the chi.bernoulli_number(k) command (for a Dirichlet character chi) by a factor of thousands.

• more details

Cylindrical Algebraic Decomposition

PEOPLE: Carl Witty

• Fixed several cases that assumed that polynomial roots were well-separated to instead do interval arithmetic with ever-increasing precision so that the correct answer will eventually be found.

Pynac Integration

PEOPLE: Mike Hansen, William Stein, Robert Bradshaw, Carl Witty

Sparse Linear Algebra

PEOPLE: Bill Hart, S. Ault

Javascript Graph Editor

PEOPLE: Rob Beezer, Mike Hansen, David Roe, Nicolas Thiery

eMPIRe Projects

PEOPLE: Bill Hart

Category Patches

PEOPLE: Nicolas Thiery, Robert Bradshaw, Mike Hansen, Craig Citro, Carl Witty

Notebook Work

PEOPLE: Mike Hansen, (William Stein), Tom Boothby

Latex for (Combinatorial) Graphs

PEOPLE: Rob Beezer, Fidel Barrera, John Palmieri, Emily Kirkman

Screenshot

Details at #5975

Wrap mpmath

PEOPLE: Fredrik Johansson

p-adic Integration

PEOPLE: Robert Bradshaw, Kiran Kedlaya

• Prepared patches to address bugs in tickets 5947, 5948; see trac 5948. These have been reviewed by Jen Balakrishnan.

• Prepared patch to address ticket 5430; see trac 5430.

p-adic Polynomials

PEOPLE: Kiran Kedlaya, David Roe

• David is preparing a large patch at trac 6048 to restructure p-adic polynomials. Besides adding new functionality, this should fix several previously reported bugs; once this is in, we should check 1210, 2943, 4656, 5075, and maybe some others.