GSoC 2018: Ideas Page

Introduction

Welcome to Sagemath's Ideas Page for GSoC 2018!

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

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

Addition of SPQR-tree to graph module

Objective:

The goal of this project is to contribute the graph theory module of SageMath with the addition of the linear time algorithm for partitioning a graph into 3-connected components [1] and the construction of the corresponding SPQR-tree [2]. Then, we will use these algorithms as pre-processing for several graph problems such as Hamiltonian Cycle, TSP, etc.

The preferred approach will be to create an interface for the OGDF library [3] that contains an efficient C++ implementation of the construction of SPQR-trees.

[1] J. E. Hopcroft, R. E. Tarjan: Dividing a Graph into Triconnected Components. SIAM J. Comput. 2(3): 135-158 (1973) [2] C. Gutwenger, P. Mutzel: A Linear Time Implementation of SPQR-Trees. Graph Drawing 2000: 77-90 [3] http://www.ogdf.net/

---

Matroid Theory (several projects)

Matroid theory in SageMath can be improved in many ways. Some potential topics, of varying difficulty, are:

• Support for signed-graphic [5], gain-graphic, frame matroids [6] (showing the graph, resigning across cuts, Whitney switching, etc.)
• Tangles and branch decompositions [4]
• Faster minor testing (finish ticket 16545 [3], and extend the ideas to non-binary matroids)
• A framework for dealing with minor-closed classes: like a set data structure, but with some support for minors.
• linear extensions/coextensions that keep track of allowed vectors.
• Bracket rings [2] /Tutte groups/universal partial fields [1] (needs: strong knowledge of algebra and Groebner bases)

[1] http://matroidunion.org/?p=2103

[2] https://en.wikipedia.org/wiki/Bracket_ring

[3] https://trac.sagemath.org/ticket/16545

[4] https://www.sciencedirect.com/science/article/pii/S0095895609000082

[5] https://arxiv.org/abs/1405.1313

[6] https://arxiv.org/abs/1609.05574

Redesigning the polynomial class hierarchy and linking with libraries

Sagemath supports univariate and multivariate polynomial arithmetic over general rings with many features. For efficiency, it is crucial that Sagemath links to low-level implementations for polynomials over certain rings, such as finite fields or integers. These things have been in Sagemath for a long time, and e.g. much of the linking code was written in early versions of Cython when it was much less powerful than it is now.

It is time to rethink and redesign the class hierarchy and the linking code for polynomials in Sagemath. The task in this project is to get an overview of the current features Sagemath supports, what we would like to do forward, and then come up with a better, cleaner design and implementation for doing this. This will entail re-linking to the many projects Sagemath already talks to, and perhaps find projects that have appeared on the OSS scene since.

Polynomial optimisation and sums of squares (multiple projects)

Optimisation problems with polynomial constraints are efficiently, in practice, solved by building an increasingly tight sequence of semidefinite programming (SDP) relaxations, one of them known as Lasserre hierarchy [1].

While Sagemath already has an ability to solve SDPs, more work has to be done in particular to implement moment matrices for polynomials and sums of squares approximations of nonnegative polynomials, and a frontend allowing to define systems of polynomial inequalities using natural syntax, similar to what already can be done with systems of linear inequalities. Another related topic would be to interface to Sagemath more SDP solvers (currently only CVXOPT is available), and possibly prototype an arbitrary precision SDP solver to avoid typical numerical difficulties arising in sums of squares-based SDP relaxations, e.g. implementing a version of the algorithm from [2].

[1] https://en.wikipedia.org/wiki/Sum-of-squares_optimization

[2] https://arxiv.org/abs/1507.03549

---

Databases of and bounds of codes (multiple projects)

Sagemath already has a considerable coding theory component. We will aim to extend it to allow constructions of optimal linear (and perhaps more general) codes, akin to a non-open-source [1]. In particular, while Sagemath is capable to compute bounds on code sizes, in many cases matching bounds of [1] (or even better than these), there is no systematic way to organise these bounds available.

The main task here will be to construct optimal (or best known) codes for given sets of parameters, in many cases combining constructions already available in Sagemath to build codes inductively. This is similar in technique to what is already done in Sagemath for Hadamard matrices and other combinatorial objects such as strongly regular graphs. Smaller tasks would be to implement linear programming bounds for constant weight codes, as well as semidefinite optimisation-based bounds due to A.Schrijver et al., see e.g. [2].

[1] http://codetables.de/

[2] https://www.math.u-bordeaux.fr/~cbachocb/articles/handbook.pdf

Implement Aspects Of Representation Theory (multiple projects)

Representation theory is a broad area of mathematics that has deep connections with many applications in other fields, such as physics and chemistry. GAP provides support for (typically finite) groups, and usually through their character theory. However, it does not contain support for standard constructions nor representations of algebras. Sagemath provides access to GAP and is currently the only software that provides systematic support for crystals, combinatorial realizations of Kac-Moody algebra representations. Sagemath also currently provides some very limited support for representation theory, but it has the framework in place to both significantly improve and increase the aspects of representation.

Some of the potential projects include:

• Implement induction and restriction functors.
• Implement methods and framework for constructing representations.
• Implement representations for Hecke algebras.
• Implement various bases for the symmetric group algebra.
• Implement group representations as modules.

• Algorithm to find invariant subspaces #11285.

• Finalize implementations of Fock space#15508, Lie algebras #14901, cellular algebras #19506, etc.

Implement Lie superalgebras

Lie superalgebras were introduced by Kac in order to unify bosons and fermions and are currently an active topic of research. The goal of this project is to extend the Lie algebra framework in Sagemath to Lie superalgebras and provide a basic implementation of the standard finite-dimensional Lie superalgebras and their representations.

Improve Root/Coxeter Systems

This project is above improving the implementation of root systems and Coxeter systems in Sagemath. It is divided into 3 main areas:

Root systems

We currently have an implementation of root systems with special inputs for finite and affine types, but there has been interest to implement an easy system for obtaining hyperbolic types. One aspect of this project is to implement such a system. Another aspect is to make the current implementation of Dynkin diagrams, Cartan matrices, and root systems more robust and generalize to, e.g., Lie superalgebras, complex reflection groups, and DAHAs.

Coxeter systems

The currently implementation of Coxeter systems currently uses the root system code and needs to be improved. In particular, this removing any ambiguity between dual types of root systems, such as types B and C.

Complex reflection groups

One last aspect is that the root system code could be used to describe a permutation representation of finite complex reflection groups, but currently this relies on the (experimental) GAP3 package. This is not strictly necessary, and we should remove this as a dependency by doing a direct implementation.

Implement Quantum Cluster (Super)Algebras

Cluster algebras and their generalizations are important algebraic objects that contain deep connections with areas such as algebraic geometry and representation theory. They have been implemented in Sagemath (see, e.g., ClusterSeed and #21254). One of the next steps is to implement the quantum version, where the variables now skew commute. (Quantum) Cluster superalgebras are a super version of (quantum) cluster algebras that are in the process of being mathematically developed, but there are currently some proposed structures. The primary goal of this project is to provide a basic implementation of known structures to help advance the field.

Rational Points on Varieties

The rational point finding on varieties/schemes is rudimentary at best in Sagemath. There are a couple simple improvements to do (such as sieving modulo primes) and some more advanced algorithms (such as Turner's thesis) for point searching to implement. Finishing #22771 for point enumeration over number fields could also be done. In all cases, these are published algorithms that need implementation in Sage.

Someone experienced with classes/inheritance could also work on aspects of the schemes code to fix issues like those in #23807. There is an issue with whether or not schemes should have a unique representation in memory which is currently inconsistently applied.

Arithmetic and Complex Dynamics

There are a number of areas that the dynamical systems code could be improved. A few specific possibilities are the following. The algorithm to determine the set of rational preperiodic points could use the following improvements:

• use the dynatomic polynomial for small periods
• allow the ambient space to be a subscheme
• allow the ambient space to be a product of projective spaces.

There are a number of other basic functionality missing from products of projective space.

It is possible some improvements to the Mandelbrot and Julia set functionality could be done. A basic implementation was done for GSoC last year so now, some other applications and/or optimizations could be done. In particular, a faster implementation through something like a divide-and-conquer approach could greatly speed up some of the slower more general functions.

There are also a number of open issues at dynamics-wiki.