Sunday, 20 November 2016

New Computer Algebra System : OSCAR

On Friday we found out the exciting news that the German funding agency (DFG) has approved a massive transregional grant here in Germany, that the computer algebra group here at the University of Kaiserslautern has been heavily involved with. There was much celebration in the Department of Mathematics where I work when we received the news!



The official announcement was made on the DFG website on Monday. If you can read German, it is here.

The grant is called "Symbolic Tools in Mathematics and their Application" and is a collaboration between multiple universities here in Germany: Aachen, Kaiserslautern and Saarbrücken, with additional external people at Siegen, Berlin, Stuttgart and Tübingen, with a total of about twenty extremely experienced PIs. The grant is initially for four years, but can be renewed up to twice, for a total of 12 years (assuming we get renewed).

The exciting news from my point of view is project B1 of the grant, which we have internally been calling project B1 : Central Software Project. This is a collaboration to produce a visionary new Open Source computer algebra system. It will be called OSCAR : Open Source Computer Algebra Resource and is to be built on four "cornerstones", namely the Gap System, Singular, Polymake and ANTIC (the internal name for a new project I've been involved with, under Claus Fieker; see below).

The Principal Investigators


Before I get to any details, in no particular order, let me introduce the PI's for the Central Software Project.


Prof. Mohamed Barakat is behind the HomAlg project. HomAlg is a GAP package that allows one to do homological algebra from a categorical point of view. (You may also know Mohamed Barakat's students Sebastian Gutsche and Sebastian Posur, who worked on HomAlg/CAP.)


Dr. Frank Lübeck is one of the main authors of the Chevie project, which is a computer algebra project for computing with generic character tables of Lie groups, Coxeter groups, Iwahori-Hecke algebras and others. It makes use of the GAP system. Over the years, Frank has been a major contributor to GAP.


Prof. Michael Joswig is one of the main authors of the Polymake system. This is an Open Source project for research into polyhedral geometry. Michael Joswig is the Einstein Professor for Discrete Mathematics/Geometry at TU Berlin.

Prof. Claus Fieker is one of the original authors of the KASH/KANT system, for algebraic number theory, which became part of Magma. He is also one of the main authors of Hecke, a new Open Source algebraic number theory project based on Nemo. (Some people may also know his postdoc, Tommy Hofmann, who is a coauthor with him of Hecke and of Nemo.)


Prof. Wolfram Decker is the coordinator of a prior Priority Project funded by the DFG on Computer Algebra, which is just concluding. Wolfram Decker is one of the people who is directly responsible for directing the Singular project and is a key member of the Algebra, Geometry and Computer Algebra group in Kaiserslautern.






In this blog, I'm only going to speak about the Central Software Project. But it is important to realise that the grant is much, much larger than this one project (it is just one of about 23 funded, interrelated projects in the grant). The Central Software Project should be thought of as providing the technical infrastructure for a very large mathematics grant in the area of computer algebra, although there are other projects within the grant that will contribute in a significant way to OSCAR. I'll maybe talk about some of them in later blogs.

The entire grant application had more than 100 pages in the preproposal and 400 pages in the full proposal, and took years to put together! At the final referee stage, the grant had 10 eminent referees, most from overseas.

What is OSCAR?


So what is OSCAR all about? Again, I can only speak about it in terms of what I understand from the PI's, and my knowledge gets greater the closer you get to the bit I will be directly involved with.

The basic idea is to develop a new Computer Algebra System, covering group theory, algebraic geometry/commutative algebra, polyhedral geometry and number theory. Our aim is to directly compete with the closed source system, Magma, hence the focus on many areas where Magma has been particularly influential.

All of the cornerstone systems (except ANTIC, which is quite new) have developed significant projects and communities in their own right, over decades, and it is our hope to combine the strengths of all these systems into a single system, which is highly integrated and easy to use. And we hope to do all this without significant disruption to those existing ecosystems. OSCAR will be a single system, which builds on all four of those existing systems and which tightly integrates those systems.

One major motivation for OSCAR is mathematical applications where it is necessary to do cross-disciplinary research, for example, where one might need a multigraded, equivariant Cox ring of a toric variety over a number field, or graphs of groups in division algebras, or matrix groups over a polynomial ring over a number field. To perform such exotic calculations, it is quite often not enough to simply interface systems such as GAP, Singular and ANTIC which separately provide either matrix groups, polynomials rings or number fields, say. One needs a very tight integration, especially if one wants such computations to be practical.

Some of us believe that it is the monolithic integration of Magma that has allowed it to be successful in the areas we hope to impact. Therefore, we in the Open Source community need to get really smart about integration in order to compete.

I personally will be working on both low and mid-level integration of the four cornerstone systems, using the programming language Julia. But there will be other efforts too, for example, there will be a coordinated serialisation effort, a parallelism project, new packages, interfaces and technologies in the cornerstone systems, and so on.

There will be a number of people other than myself hired to work on OSCAR. Let me mention two that will be associated with the Central Software Project specifically.

Dr. Reimer Behrends, who shares an office with me, and who is one of the main architects and authors of HPC-GAP and HPC-Singular will be working on aspects of parallelisation in OSCAR.

As mentioned already, Sebastian Gutsche will be working on both GAP and Polymake integration with the other cornerstones. He will be doing this using Julia, and in other ways.

How am I involved in this?


So, the rest of what I'm going to write is just about the part I have been involved with. The OSCAR website when it goes live will be the best source of information on the direction the overall project is going.

Some of you may have heard of my work on the Nemo project (joint work with Claus Fieker, Tommy Hofmann, Fredrik Johansson and others). This is a computer algebra package written in the Julia programming language. The main idea behind this, from my perspective, is to demonstrate two things:

1) It is possible to integrate multiple packages, such as Flint, Arb, Singular, Hecke, Antic (the C library -- not the ANTIC cornerstone mentioned above) and so on, in an efficient way, using the Julia programming language. And we've had some early success; already, we have demonstrated that it is possible to build Singular polynomial rings over coefficient rings provided by Nemo itself, or by Antic or Hecke, etc.

2) It is possible to develop extremely efficient implementations of generic algorithms over generic rings (rather than the specific rings that we have highly optimised implementations for, e.g. in the Flint system). This is possible in the Julia language, due to its innovative type system and JIT (Just-In-Time) compiler.

(It should be noted that Polymake has a very sophisticated form of runtime specialisation, and what may be thought of as an early kind of JIT compiler.)

I personally believe that high level algorithms implemented in an old-style interpreter are sometimes not enough to get a practical implementation for some research questions. Sometimes, the difference between a practical implementation and failure, is not bottlenecks in low level C code, but generic algorithms implemented in a high level, interpreted language. But unfortunately, we don't have enough skilled, low-level developers to reimplement such things in a low-level language every time they appear!

So one of the things I am particularly interested in, is applications where a "mid-level" language like Julia can be used to speed up generic algorithms. This has been one of the ideas behind my work on Nemo, so far, and the reason why I chose to develop it in the Julia programming language. Julia facilitates high-level constructs, with low-level performance.

Already, on top of Nemo, Prof. Claus Fieker and Dr. Tommy Hofmann, have built the Hecke system, also written in the Julia programming language, for doing computations in orders of number fields and class group computations. This is a culmination of efforts of Claus Fieker's project under the Priority Project, which I mentioned above.

What is the ANTIC cornerstone?


So with all of that background in mind, I can now describe the ANTIC cornerstone of the Central Software Project. ANTIC is the internal name we have had for the combination of GMP/MPIR, MPFR, Flint, Antic, Arb, Nemo and Hecke, that we have been busy with integrating and in some cases, developing under the Priority Programme. It will now become one of the four cornerstones of the Central Software Project, and will essentially be responsible for providing algebraic number fields and related things to the new Computer Algebra System, OSCAR.

We now begin the task of doing our part to help with the integration of ANTIC with the other cornerstones of the OSCAR system, namely Singular, Polymake and GAP.

There are already preexisting and independent projects underway to integrate Flint and Antic (the C library) into GAP as packages, which we aim to support in whatever way we can.

And within our collaboration, Michael Joswig has already suggested projects to me that will require integration of Antic with Polymake, and work is already quite advanced in the direction of integrating Singular and Nemo/Hecke. And there have already been extensive discussions about future integration of ANTIC with GAP, probably via Nemo and/or Hecke.

The ANTIC project achieves its integration, and some of its generic algorithms are directly implemented in, the Julia programming language. We aim to contribute the expertise we have been developing here in Kaiserslautern, in the Julia language, to the wider integration of the four cornerstones in OSCAR.

Will you join us?


Obviously OSCAR is a massive undertaking. We hope that the mathematics community and Computer Algebra communities at large will support our efforts.

OSCAR is an Open Source project, and we hope that it will attract many contributors over the years and that it will support many Masters and PhD projects!

The really important thing is that OSCAR is motivated by a large number of serious mathematical projects across computer algebra. These applications will drive development of the project and I believe, lead to a high quality, well-tested project, leading to many substantial contributions and publications for a long time to come.

Join us if you are interested. In coming days we will very quickly be setting up websites and infrastructure. In the meantime, feel free to contact me or any of the other people you know who are involved with this project.

And join us in celebrating what is a fantastic investment in Open Source computer algebra, that we hope will be of great benefit for the mathematical community at large!

Further Information


I hope to keep people up-to-date with what I'm personally working on, through this blog. I'll also link to other press releases, blogs and articles about OSCAR and our grant, as I become aware of them. So follow this blog if you are interested in hearing more as things develop.

Here is the University of Kaiserslautern press release (German).

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