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I spent a week in St Andrews working with Markus Pfeiffer on an ongoing project which we are working on writing and implementing an algorithm to construct 2-closed Majorana representations. Funding for this trip was provided by CoDiMa.
Majorana algebras are non-associate algebras used to study the Monster group and its subgroups. They can be studied either in their own right, or as Majorana representations of certain groups. Many examples of Majorana algebras have been constructed by hand but it has become clear that, in order to construct bigger and more interesting algebras, a more computational approach is needed.
In a celebrated paper in 2012, Akos Seress announced the existence of an algorithm to constuct 2-closed Majorana representations. Sadly Seress passed away in 2013 and the full details of his algorithm and his results were never published. Recovering his work has been an important aim of the theory ever since.
This was my second visit funded by CoDiMa to work on this project. After the first visit, we had implemented such an algorithm and had started to reproduce Seress’ results. However, constructing Majorana representations is very expensive both in terms of time and memory and so the algorithm requires a lot of optimisation, which was the focus of our work this time around.
The week was a sucess and we have now completely recovered Seress’ results and we are expecting to shortly be able to construct representations larger than those achieved by Seress.
Our work is available on GitHub.
The 8th International Workshop on Parallel Symbolic Computation (PASCO’17) took place in Kaiserslautern, Germany, July 23-24, 2017, and was co-located with the International Symposium on Symbolic and Algebraic Computation (ISSAC’17). This instance of PASCO continued the mission of promoting and advancing parallel algorithms and software in all areas of mathematical computation. It therefore occupies a unique space in the landscape of international conferences and workshops, that brings together (parallel) systems developers and theoreticians in the broad area of mathematical software, in particular for symbolic computation. As in previous instances, PASCO’17 was generously supported by Maplesoft, who donated 2 Maple licences for the recipients of the best paper and the best student paper awards in the programme of PASCO’17. Details about the event can be found in the preface to the PASCO’17 proceedings, published in the ACM Digital Library: https://dl.acm.org/citation.cfm?id=3115936&picked=prox
The travel funding by Codima allowed me (Hans-Wolfgang Loidl) to attend this event, as General Chair for PASCO’17, in person and to oversee the running of the workshop. I was particularly pleased by the good attendance for the sessions of this event: the workshop programme contained a total of 13 contributed papers, reflecting the focused nature of this workshop, but attendance to the sessions was in the range of 30-40 academics and students, as well as representatives from Maplesoft and other companies. This reflects the wider interest in the high-performance aspects of this field.
From a Codima point of view, this was a good opportunity to flag the presence of this UK network and to strengthen its international connections. Codima members have been involved in the running of this event as General and Publicity chairs, and continue to take a leading role in the growing of this community. New links have been established between Codima members and the invited speakers from PASCO’17: the group in Glasgow is in collaboration with Prof Florent Hivert, on parallel symbolic algorithms in the area of semigroup enumeration. We hope to further firm up these links by inviting Prof Hivert for a research visit to Scotland.
Hans-Wolfgang Loidl (Heriot-Watt)
The CoDiMa project supported my participation in “All Kinds of Mathematics Remind me of You: Conference to celebrate the 70th Anniversary of Peter J. Cameron“, held at the University of Lisbon, 24-27 July 2017. This conference brought together many colleagues, students and collaborators of Peter Cameron, and a diverse range of interesting mathematical research (covering Peter’s diverse range of interests) was presented and discussed.
This conference gave me the opportunity to present and discuss my recent algorithms and programs (in GAP/GRAPE) to exploit graph symmetry in graph colouring, in particular in the difficult problem of computing the chromatic number of a graph. I was also able to discuss the application of these programs to the determination of the “non-synchronizing” primitive permutation groups (of interest to researchers in semigroup and automata theory) of degree at most 255.
In June 2017 Alexander Konovalov took part in the conference “Groups, Rings and the Yang-Baxter equation” (Spa, Belgium). He gave a talk “GAP Group Rings Toolkit” with an overview of the functionality to work with group rings available in GAP and four of its packages, demonstrated the Jupyter kernel for GAP, and organised a coding sprint to work on the Wedderga package. As a result, Wedderga development version has been migrated from Bitbucket to GitHub (https://github.com/gap-packages/wedderga), and a new collaborator, Dr Sugandha Maheshwary (ISER Mohali), had submitted her first pull request to Wedderga.
After meeting at the Workshop on Permutation Groups: Methods and Applications in Bielefeld in Germany, Markus Pfeiffer became interested in some computational work which I am developing as part of my PhD. He kindly invited me to St Andrews to spend a week working together on my code. Funding for this trip was provided by CoDiMa.
The work concerns developing and implementing an algorithm which can construct Majorana algebras, objects which occur in the study of the Monster group and its associated representation, the Griess algebra. In particular, I am interested in studying these algebras as Majorana representations of certain finite groups.
The algorithm takes as its input a finite group and a generating set of involutions. It considers all possible Majorana representations of the group with respect to the generating set and then, for each representation, either attempts to construct it or shows that it cannot exist.
In 2012, Akos Seress announced that he had constructed such an algorithm and published a list of groups whose Majorana representations he had been able to classify. However, Seress sadly passed away before he was able to publish the details of either his algorithm or of the representations which he had constructed. Reproducing his work has been an important aim of Majorana theory ever since.
The code is currently able to construct the Majorana representations of some groups, but we have not been able to reproduce the full results of Seress’ work. Together, we have been working on improving the methods in the algorithm to extend its capabilities. Improvements can come either from better implementation of the current methods, or from finding new approaches from theoretical work on the algebras.
This work is of particular interest as these algebras are defined over the reals and their construction involves some linear algebra over rational numbers. Improving GAP’s functionality over fields of characteristic zero is something which is being actively worked on and will benefit this problem as well as many others.
I also got the opportunity to present my work at the School of Mathematics and Statistics’ Pure Colloquium.
Overall, the week was very productive and we look forward to working together in the future.
Yet again I attended the Nikolauskonferenz in Aachen this year, funded by CoDiMa.
At the meeting Chris Jefferson and I presented our work with Rebecca Waldecker, and co-funded by CoDiMa, on search and canonical images in permutation groups. A recent submission can be found here, and a further one is coming out soon.
Another notable talk was given by Mikaël Cavallin from Kaiserslautern: He and Donna Testerman found a bug in a paper by Seitz from 1987 which is widely used in algebraic groups. This reminded me of our CoDiMa event in January, where Carmen Rovi visited us to learn about how GAP computes Schur multipliers, and we suspected that there was a bug in GAP, but it turned out to be a bug in an old paper.
Richard Parker and I met mainly at breakfast and discussed high performance low level algorithms such as his meataxe64, or multiplying permutations on millions of points, making full use of modern computer systems, which according to Richard, humanity is too stupid to program.
Two further talks that caught my attention were Imke Toborg’s talk on An Algebraic View on a Composite Functional Equation on Groups, because I first thought: why would you do that? and then: actually this is really interesting, and Julian Brough’s talk about Central Intersections of Element Centralisers, because I like this kind of group theory.
Of course all the other talks were interesting too, and I very much enjoyed being in Aachen again meeting everyone and doing research – Cambridge style! once more. A special thank you goes to Frank Lübeck for organising the event. I hope to see everyone back in Aachen next year!
In October 2016, Markus Pfeiffer visited Alan Logan in Glasgow to work on the practical implementations of methods to compute with hyperbolic groups. This is an important direction of research, and it was once more highlighted at the “Computation in geometric and combinatorial group theory” workshop at the ICMS in Edinburgh in July this year that we need to have a usable and publicly available GAP implementation of these. The progress report on this visit is available here. The work on the new GAP package continues…
Just like every year since about 2004 I have attended the Nikolauskonferenz in Aachen in December 2015.
Nikolaus is a relatively small meeting of mathematicians interested in group- or representation theory, and computational methods in these fields in particular. As such this meeting is a good venue to meet users of the GAP system, hear about their experiences and do some advertising work for HPC-GAP.
One particular highlight to be mentioned here was my conversation with Sergio Siccha, who just started his PhD in Aachen and wants to use HPC-GAP, and Jürgen Müller, one of the authors of the Orb GAP package. Sergio is going to attend our first joint GAP and SageMath days in January and we will work on a HPC version of the orbit-by-suborbit algorithm.
Thanks to Frank Lübeck, who has been organising this meeting for as long as I can remember, to all the speakers who gave interesting talks, and all attendees who made this meeting a memorable experience.
Last but not least, thanks to CoDiMa for making this visit possible for me!
In our discussions we found out that search in permutation groups is a topic that intersects our interests, and Rebecca suggested that we should apply for a grant from Deutsche Forschungsgesellschaft (DFG) to fund mutual visits over the course of a year to continue our collaboration.
Rebecca’s application has been successful and we will receive 6460 EUR in travel support for two visits by Rebecca Waldecker to St Andrews and two visits by Chris Jefferson and Markus Pfeiffer to Halle.
We are planning to attack the theory and practice of search in permutation groups such as group intersection, automorphisms of structures, canonical images and parallelisation.
With our different backgrounds in group theory and computation, we would also like to make the theory more accessible to programmers, and the practical aspects more accessible to mathematicians.
What makes this topic even more exciting and promising is Babai’s most recent publication on the complexity of the graph isomorphism problem.
Prof. Rebecca Waldecker visited St Andrews on August 23rd-29th, 2015. From the report by Markus Pfeiffer, “… During her one-week visit to St Andrews, Rebecca Waldecker, Chris Jefferson, and I discussed many ideas around the notion of fixity, the partition backtrack algorithm first published by Leon and the canonical image algorithm. We found it to be a very productive week and agreed to apply to a further grant to make our work into a longer-term project.”