- Markus Pfeiffer (School of Computer Science, St Andrews, UK)
- Florian Rabe (Jacobs University, Bremen, Germany)
- Nicolas Thiéry (Université Paris Sud, France)

There is a wealth of mathematical software available today, most of which are free and open-source, but very specialised: for example nauty for graph isomorphism, Singular for polynomials, Pari for number theory. Additionally there are databases of objects like the OEIS, or the Small Group Library.

In contemporary mathematics research, we use and combine these specialised tools, which requires domain-specific knowledge, data access and conversion, and low-level programming for interfacing between them.

This is tedious, error-prone, and does not scale: Addressing the demand for better composability there are huge software packages like SageMath, data description languages like OpenMath, and RPC mechanisms like SCSCP.

The aim of this session is to provide a forum for developers and users of mathematical software with an interest in composablity and interoperability of, and knowledge and data exchange between systems, to share experiences, solutions, and a vision for the future.

OSCAR (Open Source Computer Algebra Resource) is a new project to build a visionary computer algebra system integrating GAP, Singular, polymake, and Antic by using the Julia programming language.

The GAP package JuliaInterface and the Julia package Gap.jl provide the low-level part of integrating GAP and Julia. They combine several features, including

- accessing all functionality in Julia from GAP,
- accessing all functionality in GAP from Julia,
- and conversion of basic types between GAP and Julia.

Unlike interfaces, the integration aims providing compatibility at the lowest level, trying to make the wall separating the two systems as low as possible. To achieve this, several technical difficulties had to be tackled, of which some will be presented in the talk.

Together with the integration of Singular and polymake into Julia (also part of the OSCAR project), JuliaInterface and Gap.jl will be used to provide integration of GAP with Singular and polymake.

Because of the low penalty when switching between the GAP and the Julia layer, this integration enables one to write algorithms combining the computational power of all four systems.

In this regard, JuliaInterface will be used to replace incomplete interface from GAP to polymake and the outdated interface from GAP to Singular, using as few binding code as possible to transparently access all foreign functionality.

Furthermore, the highly efficient integration of Julia into GAP allows one to implement fast versions of algorithms for GAP in Julia instead of C.

This is joint work with Thomas Breuer.

In 2004, I started the Sage software project to create, for me, a viable alternative to Magma for my number theory teaching and research. It has grown much larger in scope and contributors over the last 14 years. I will describe the motivation behind Sage, what technology has come out of the project over the years (such as Cython and web-based code notebook interfaces), and where Sage might go when we have the resources that we need.

There is a large ecosystem of mathematical software systems. Individually, these are optimized for particular domains and functionalities, and together they cover many needs of practical and theoretical mathematics. However, each system specializes on one particular area, and it remains very difficult to solve problems that need to involve multiple systems. Some integrations exist, but the are ad-hoc and have scalability and maintainability issues. In particular, there is not yet an interoperability layer that combines the various systems into a virtual research environment (VRE) for mathematics.

The OpenDreamKit project aims at building a toolkit for such VREs. It suggests using a central system-agnostic formalization of mathematics (Math-in-the-Middle, MitM) as the needed interoperability layer. In this talk, we report on a case study that instantiates the MitM paradigm the systems GAP, Sage, LMFDB and Singular to perform computation in group and ring theory.

Our work involves massive practical efforts, including a novel formalization of computational group theory, improvements to the involved software systems, and a novel mediating system that sits at the center of a star-shaped integration layout between mathematical software systems.

In Computational Mathematics, Computer Algebra and Proof systems have grown up in parallel with little crossover. Two approaches seem to dominate their combination. There have been efforts to prove individual algorithms, such as Groebner Basis, inside a Proof system. There have been efforts to communicate between systems where the Computer Algebra system is used as an untrusted Oracle for the Proof system. Could the Computer Algebra system be made trustworthy?

Typeclasses have three components: Signatures, Carriers, and Propositions. Axiom assigns Signatures through its “Category” hierarchy. It assigns Carriers, known as the Representation through its “Domain”. This partition defines an API but allows multiple representations such as sparse and dense polynomials.

The missing component is Propositions. This research effort associates Propositions with both the Category and Domain components. The goal is to collect both Category-level and Domain-level Propositions and use them to prove algorithms ‘in-situ’. In Axiom, GCD occurs in 22 different Domains, each of which has different Propositions available. In addition, the collected Propositions define properties for user-level operations, e.g. commutitivity.

Algorithms proven correct would allow Axiom to be a Trusted Oracle.

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