GSoC 2022: Ideas Page

Introduction

Welcome to SageMath's Ideas Page for GSoC 2022! (Last year 2021)

SageMath's GSoC organization homepage -- the hub for submitting applications and where the everything on Google's side is organized. (Timeline)

Please subscribe to the sage-gsoc mailing list and the GAP developer list for discussion on possible GAP GSoC projects. Also, make sure you have gone through the information regarding application procedures, requirements and general advice. The Application Template is also available on that wiki page. Archives of past GSoC project ideas can be found here.

All projects will start with an introduction phase to learn about SageMath’s (or sister projects') internal organization and to get used to their established development process. We also require you to show us that you are able to execute actual development by submitting a relevant patch and/or reviewing a ticket via Trac of the project you are interested in applying to. The developer guide is a great comprehensive resource that can guide you through your first steps in contributing to SageMath.

Apart from the project ideas listed below, there is also a comprehensive list of future feature wishes in our trac issue tracker. They might contain the perfect project idea for you we didn't even think about!

Project Ideas

Here is a list of project proposals with identified mentors. Other well-motivated proposals from students involving SageMath in a substantial way will be gladly considered, as well.

Improve (free) module implementations

Mentor

Travis Scrimshaw

Area

Linear Algebra, Performance, Refactoring

Skills

Understanding of linear algebra and object-oriented programming. Cython experience is highly recommended.

Length

175 hours

Difficulty

Medium-easy

SageMath has multiple implementations of free modules:

1. Finite dimensional coordinate representations in the "standard" basis using FreeModule that provides both a dense and sparse implementation. 2. Using CombinatorialFreeModule (CFM) as (possibly infinite dimensional) sparse vectors.

There are various benefits to each implementation. However, they are largely disjoint and would mutually benefit from having a common base classes. In particular, having a dense implementation for CFM elements for applications that require heavier use of (dense) linear algebra. The goal of this project is to refactor these classes to bring them closer together (although they will likely remain separate as they are likely not fully compatible implementations for the parents).

Rewrite exterior algebra and implement Gröbner bases

Mentor

Travis Scrimshaw, Vic Reiner

Area

Algebra, Performance

Skills

Understanding of abstract algebra and Cython. Knowledge of Gröbner basis is recommended.

Length

175 hour and 350 hour variants

Difficulty

Medium-hard

The exterior (or Grassmann) algebra is a fundamental object in mathematics, in particular with applications to physics and geometry. It could be considered as the closest non-commutative analog of polynomials where the variables skew-commute with each other. The current implementation uses a basis indexed by subsets (as tuples), but a more efficient version would be indexed by integers encoding membership by the binary string. The first goal is to do this change (#32369). The second goal of this project would be to implement an algorithm for Gröbner basis for their ideals in order to construct quotient algebras. A variation of this project would be to improve the implementation of commutative graded algebras to not rely on the more generic plural for computations (except perhaps those involving ideals). For the ambitious, these computations would be extracted to an independent C++ library for many common rings (implemented using other libraries).

Implement Schubert and Grothendieck polynomials

Mentor

Travis Scrimshaw

Area

Algebra, Combinatorics, Schubert Calculus

Skills

Foundations in combinatorics, experience reading research papers.

Length

175 hours

Difficulty

Medium-hard

Schubert calculus can roughly be stated as the study of the intersections of lines, through which certain algebras arise that can be represented using Schubert polynomials and Grothendieck polynomials. The main goal of this project is to finish the implementation started in #6629, as well as implement the symmetric Grothendieck polynomials and their duals in symmetric functions.

Tensor operations in Sage using Python libraries as backends

Mentor

Matthias Koeppe

Area

Linear/multilinear algebra

Skills

Solid knowledge of linear algebra, Python experience, ideally experience with numpy, PyTorch, JAX, or TensorFlow

Length

350 hours

Difficulty

Hard

In this project, we develop new backends for the tensor modules from the SageManifolds project. Amongst the goals of the project are such elements as a fast implementation of tensor operations using numpy and using TensorFlow Core and PyTorch.

Enhanced optimization solver interfaces for Sage

Mentor

Matthias Koeppe

Area

Optimization

Skills

Solid knowledge of linear optimization, Python experience, ideally experience with Python optimization interfaces

Length

350 hours

Difficulty

Hard

See Meta-ticket #26511: Use Python optimization interfaces: cvxpy, PuLP, Pyomo, cylp...

Fast evaluation of symbolic expressions

Mentor

Vincent Delecroix and Isuru Fernando

Area

Symbolic expression

Skills

Basic math background in algebra and analysis. Fluent in Python and C/C++. Knowledge of compilers, Cython, or parallelization (openmp) would be useful.

Length

350 hours

Difficulty

Medium

The simplest example of a function is given by univariate polynomials such as P(x) = x^3 - 2*x + 3. A more complex function is F: (x, y) -> (cos(sqrt(x) + 1) * sin(y), tan(x^2 + 2) + y). For this project we are interested in making the evaluation of such expression at given concrete values fast and reliable. SageMath already has a "compiler" for symbolic expression that is used through fast_callable

sage: x, y = SR.var('x,y')
sage: f = cos(sqrt(x) + 1) * sin(y)
sage: g = fast_callable(f, vars=[x,y], domain=float)
sage: g(2.3, 3.5)
0.2844686555174862

The objective of this project is to rewrite the code for fast callable using the more modern project symengine. Doing so the applicant is likely to contribute to both symengine and SageMath projects. Here is a potential list of subtasks

  1. make sage code compatible with symengine

    • Some algorithms in sage do use fast_callable (such as integration routine, numerical differential equations solver, computation of geodesics on manifolds, ...). For each of them, one need to check whether the symengine functions can be used as a drop-in replacement and possibly adapt the sage code. Documentation and tests should be adapted accordingly.

  2. add support for more data types in symengine (in order to support more sage types)

    • Making numpy arrays support more types such as
      • integers (GMP mpz_t and flint fmpz_t)

      • rationals (GMP mpq_t and flint fmpq_t)

      • real numbers (MPFR mpfr_t)

      • intervals (MPFI mpfi_t) and balls (arb arb_t and acb_t)

      An example of such implementation are the quaternions.

  3. optimization

    • Time benchmark fast_callable against symengine

    • Analyzing and optimizing accuracy (for floating points numbers). For example the order of operations do matter for accuracy.
    • Analyzing and optimizing the size of the evaluation tree: given an expression there are plenty of way to evaluate it. For example P(x, y) = x^4 + 2*x^2*y + 3*x^2 + 2*x*y + 2*y^2 + 2*y + 1 can be evaluated naively. But we can also rewrite it as P(x,y) = (x^2 + y + 1)^2 + (x + y)^2 and get another evaluation scheme. This problem of determining the optimal evaluation tree is known to be NP-complete.

    • Generate code for multiple input arrays so that compiler can optimize it better.
  4. parallelization

Related projects

Edge connectivity and edge disjoint spanning trees in digraphs

Mentor

David Coudert

Area

Graph theory

Skills

Solid knowledge of graph theory and graph algorithms, Python and C/C++ experience

Length

350 hours

Difficulty

Hard

The current method used for finding edge disjoint spanning trees in directed graphs (digraphs) relies on mixed integer linear programming and it may fail on some instances (see ticket #32169). The problem has been fixed for undirected graphs, implementing a combinatorial algorithm (see ticket #32911). The goal of this project is to implement combinatorial algorithms for finding edge disjoint spanning trees in digraphs. We will particularly consider the following algorithms:

Improve Height Functionality

Mentor

Ben Hutz and Alex Galarraga

Area

Algebraic Geometry

Skills

Python experience, abstract algebra, basic algebraic geometry, number theory

Length

175 hours

Difficulty

Medium Hard

There are some issues with the current implementations of heights in Sage

Related to #32687 is implementing the more efficient algorithm from Krumm from the paper mentioned in that ticket.

Implementation of generalizations of RSK algorithm

Mentor

Tomohiro Sasamoto

Area

Combinatorics, Probability Theory

Skills

Python experience, understand combinatorial algorithms, experience reading research papers

Length

175 hours

Difficulty

Medium Hard

In Sage, the classical RSK algorithm is implemented. Such algorithm admits extensions as Sagan and Stanley's version, which puts in bijection pairs of skew tableaux with matrices of sequences. Another extension of the RSK is given by a recent algorithm by Imamura, Mucciconi and Sasamoto.

As these generalizations of the RSK are not yet available in Sage we aim to implement them. The project will include creating new combinatorial operations on tableaux and new realization of Kashiwara operators. We also aim at creating Sage functions with which one can visualize such new operations.

Implement piecewise functions of one or several variables

Mentor

Yuan Zhou

Area

Geometry

Skills

Knowledge of linear algebra and polyhedral geometry, Python experience.

Length

175 hour and 350 hour variants

Difficulty

Medium

See Meta-ticket #20877: Piecewise functions, polyhedral complexes, piecewise functions of several variables, periodic piecewise functions

Make polyhedral algorithms verifiable

Mentor

Yuan Zhou

Area

Geometry

Skills

Knowledge of polyhedral geometry and linear programming, Python experience.

Length

350 hour

Difficulty

Medium-hard

See Meta-ticket #31343: Certified polyhedral computation

GSoC/2022 (last edited 2022-04-05 13:48:07 by dcoudert)