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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.”
Dima Pasechnik’s visit to Birmingham to collaborate with Sergey Shpectorov took place from 22/07/2015 to 24/07/2015. This is a quote from their report on this visit: “The focus of this collaboration is on the improvements to Sergey’s GAP package for computation of axial algebras. These are commutative non-associative algebras generated by idempotents satisfying certain fusion rules. The motivating example is the Griess-Norton algebra for the Monster sporadic simple group. The available programs for computations with axial algebras rely on the linear algebra functions available in GAP. This means that the program cannot handle partial algebras (intermediate stages) over the rationals whose dimension exceeds two-three thousand. At the same time, the algebras that we construct have significant groups of symmetries and, in particular, every partial algebra is a G-module for a certain group G. It is hoped that an efficient implementation of the basic module operations using the known character table of G will allow to increase the limit on the dimension by several orders. It is also hoped that, due to the fundamental nature of the module operations, the new routines will also find applications in many other places.”
“Two principal aims of my visit to St Andrews were to get familiar with package design and developing a parallel implementation of existing sequential code. Secondary objectives included discussing the upcoming software carpentry workshop and GAP days in Manchester in November, GAP’s built-in and user-contributed profiling features and creating new objects and methods for them.
We focused on the *philosophy* of the package design good-practice as is usually practiced in the GAP community, and Alexander said this will appear shortly as a blog post on his website; I’ve volunteered to help develop a hands-on tutorial in creating a package using the “example” package as a template. As far as parallel computing goes, we studied the shared-memory model implemented in HPC-GAP and the distributed model that SCSCP package is built around, and implemented some of my code in both so I have now a working example in each.
My secondary objectives were all happily satisfied at least in part. The one thing that we didn’t cover fully was profiling, and Alexander suggested that perhaps this was a sensible subject for a tutorial at the upcoming GAP days.”
Our short research visits programme successfully continues. Since the beginning of the project, we have received 14 applications for travel support, and 12 visits already took place. This is an overview of summer travel activities which were fully or partially funded by CoDiMa after the publication of the previous travel update.
In June, (more…)
We have already funded several trips under our short research visits programme.
Mark Kambites (Manchester) visited St Andrews on 21-23 April 2015 for the initial planning meeting and discussing further collaboration on implementing a tropical algebra/geometry package for GAP, based on his earlier work with Markus Pfeiffer (St Andrews).
Alexander Konovalov visited Newcastle University on May 21st, 2015 to meet the members of the algebra research group in Newcastle: (more…)