About the OSCAR project

The OSCAR project uses the Julia programming language to integrate the four computer algebra systems GAP, polymake, Singular, and Antic (Hecke, Nemo) developed within the TRR 195 into a visionary next generation open source computer algebra system surpassing the combined mathematical capabilities of the underlying systems.

OSCAR allows users to construct new mathematical objects efficiently by combining existing building blocks from any of the cornerstones and equip these objects with mathematical capabilities exceeding those of the individual systems in a transparent way.

The four cornerstones

This project builds on four leading computer algebra systems which are, with the exception of the new system ANTIC, widely used internationally. Although the new computer algebra system tightly integrates the four cornerstones, the individual systems will continue to be developed further. This includes working together with the developers of the individual cornerstones to develop the cornerstones in a way the best integration can be achieved.

Nemo, Hecke are evolving number theoretic software projects focusing on computations in and with number fields and generic finitely presented rings. ANTIC is written in a mixture of C and Julia, building on top of the very successful FLINT project of William Hart, while Hecke is pure Julia. In order to reach the mathematical goals of the OSCAR project, the functionality of both will be considerably extended.

GAP is a system for computational discrete algebra, with particular emphasis on group and representation theory. One of the distinguishing features of GAP is that it provides a general purpose high level interpreted programming language especially suited to algebraic applications. It is easy to extend GAP’s functionality; currently over 120 GAP packages contributed by third party authors are distributed with each release. GAP is integrated into OSCAR via the GAP.jl Julia package.

The system polymake is a standard tool for dealing with convex polytopes, polyhedral fans, tropical hypersurfaces, and related objects from combinatorics and geometry. It is designed as a hybrid written in C++ and Perl. A sophisticated rule based mechanism decides which low-level C++ functions are to be called to satisfy user demands. Polymake is integrated into OSCAR via the Polymake.jl Julia package.

Singular is a well-established computer algebra system for polynomial computations, with particular emphasis on applications in algebraic geometry, commutative algebra, and singularity theory, and with two subsystems for non-commutative algebra, PLURAL and LETTERPLACE. Singular has a kernel written in C++ and provides its own interpreted user language. Singular is integrated into OSCAR via the Singular.jl Julia package.

The role of Julia

Julia is a an actively developed interactive and expressive programming language. Julia offers:

• data types that can depend on other types, for example modeling matrices over polynomial rings over number fields;
• Just-in-time compilation (JIT) to compile code at runtime to retain efficient performance;
• capabilities of accessing low-level C/C++ data structures in the kernels of the cornerstones efficiently;
• easy access to standard third-party libraries, (e.g., for string manipulation, visualisation, networking).

Julia serves as an integration layer allowing the four cornerstones to communicate in a more direct way than through unidirectional interfaces. Furthermore it serves as high-level language for implementing efficient algorithms utilizing all cornerstones.