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 sagegsoc 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!
Contents
 GSoC 2022: Ideas Page

Project Ideas
 Improve (free) module implementations
 Rewrite exterior algebra and implement Gröbner bases
 Implement Schubert and Grothendieck polynomials
 Tensor operations in Sage using Python libraries as backends
 Enhanced optimization solver interfaces for Sage
 Fast evaluation of symbolic expressions
 Edge connectivity and edge disjoint spanning trees in digraphs
 Improve Height Functionality
 Implementation of generalizations of RSK algorithm
 Implement piecewise functions of one or several variables
 Make polyhedral algorithms verifiable
Project Ideas
Here is a list of project proposals with identified mentors. Other wellmotivated 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 objectoriented programming. Cython experience is highly recommended. 
Length 
175 hours 
Difficulty 
Mediumeasy 
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 
Mediumhard 
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 noncommutative analog of polynomials where the variables skewcommute 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 
Mediumhard 
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 Metaticket #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
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 dropin replacement and possibly adapt the sage code. Documentation and tests should be adapted accordingly.
add support for more data types in symengine (in order to support more sage types)
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 NPcomplete.
 Generate code for multiple input arrays so that compiler can optimize it better.
parallelization
Related projects
The (now abandoned) project Theano might be of interest for optimization.
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:
Harold N. Gabow: A Matroid Approach to Finding Edge Connectivity and Packing Arborescences. J. Comput. Syst. Sci. 50(2): 259273 (1995) https://doi.org/10.1006/jcss.1995.1022
Anand Bhalgat, Ramesh Hariharan, Telikepalli Kavitha and Debmalya Panigrahi: Fast edge splitting and Edmonds' arborescence construction for unweighted graphs. ACMSIAM symposium on Discrete algorithms (SODA), pp 455464, 2008 https://users.cs.duke.edu/~debmalya/papers/soda08splitting.pdf
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
#32687 error in height difference bound
#32686 points_of_bounded_height for projective space is incorrect
A very advanced student could also finish #21129 ArakelovZhang pairing of rational maps
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 
Make polyhedral algorithms verifiable
Mentor 
Yuan Zhou 
Area 
Geometry 
Skills 
Knowledge of polyhedral geometry and linear programming, Python experience. 
Length 
350 hour 
Difficulty 
Mediumhard 

TEMPLATE
Project Title
Mentor
Name(s)
Area
Mathematical and/or technical scope ...
Skills
What the student should bring ...
Length
Mediumterm or longterm
Difficulty
Easy, medium, hard, etc.
...