Algorithms for black box groups

CoDiMa partially supported the visit of Prof. Alexandre Borovik (University of Manchester) to Dr Sukru Yalcinkaya (Istanbul University) on 8-21 September 2018. They reported the following:

In this short period of very intensive work in a small group (A. Borovik, S. Yalcinkaya (Istanbul University), and M. Lavrauw (Sabanci University)) we achieved a significant further progress towards efficient algorithms for analysis of black box groups representing classical linear groups over finite fields of odd characteristic. These results are highly significant for probabilistic methods of computational group theory. Unlike much of the previously existed methods which were conditional on the existence of Discrete Logarithm Oracles, our algorithms run in time polynomial in input length, work for very large sizes of fields without use of oracles of any kind and have been successfully tested for prime fields of order about 10^70.

GAP Jupyter interface included in GAP 4.9.2 release

The new minor release of GAP, version 4.9.2, is now available for download from the GAP website at https://www.gap-system.org/Releases/. It includes the new JupyterKernel package by Markus Pfeiffer which provides a so-called kernel for the Jupyter interactive document system. This package requires Jupyter to be installed on your system (see instructions here). It also requires GAP packages IO, ZeroMQInterface, json, and also two new packages by Markus Pfeiffer called crypting and uuid, all included into GAP 4.9.2 distribution. The JupyterKernel package is not yet usable on Windows.

The complete description of changes introduced in this release, with links to the documentation and to GitHub pull requests is available here. Please also see the release announcement in the GAP Forum.

Computations in small overlap monoids

For several days in June 2018, Mark Kambites (Manchester) visited James Mitchell in St Andrews, supported by the CoDiMa research visits programme. The visit allowed Mark Kambites and James Mitchell to discuss algorithms for performing computations in small overlap monoids. Efficient algorithms for various problems in this important class of finitely presented monoids were developed by Mark in a series of papers published between 2007 and 2011. The most basic of these (to solve the word problem) was implemented by Mark himself in the GAP SmallOverlap package in 2008. James and students have recently re-implemented this much more efficiently, and also begun to implement Mark’s other algorithms for more sophisticated tasks such as computing normal forms. However, they have encountered some problems with practical implementation of the algorithms in Mark’s papers. The visit allowed James to explain these issues, and Mark to clarify how some of the algorithms were supposed to function. In addition, the visit allowed Mark to attend the 28th NBSAN meeting, which provided a valuable opportunity for discussions with other experts in the field.

Symmetry vs Regularity: The first 50 years since Weisfeiler-Leman stabilization

Madeleine Whybrow reports:

With the support of CoDiMa, I attended the conference “Symmetry vs Regularity: The first 50 years since Weisfeiler-Leman stabilization” in Pilsen, Czech Republic. It was a very impressive event and nearly all of the major contributors to the field of algebraic graph theory over the past 50 years were present. There were some great talks presenting both the history of the area and also the new directions of research.

There was a strong computational theme throughout the conference and there were many stimulating talks and conversations on this topic. I gave a talk in which I presented my work with Markus Pfeiffer developing an algorithm in GAP to construct Majorana representations. This was well received and many people were very interested in our work, particularly in the methods that we use. Overall, it was an interesting and enjoyable week and I had the chance to meet lots of people and have lots of useful discussions.

GAP Tutorial at PGTC 2018 in St Andrews

The 20th Postrgaduate Group Theory Conference (PGTC) takes place in St Andrews on 17th-19th July 2018 (Tuesday-Thursday). We organise a satellite hands-on tutorial on the computational algebra system GAP for PGTC participants on Monday July 16th and Friday July 20th. See the tutorial page for further details.

New major GAP release: announcing GAP 4.9.1

The new major release of GAP, version 4.9.1 release, is now available for download from the GAP website at https://www.gap-system.org/Releases/. The complete description of these and other changes, with links to the documentation and to GitHub pull requests is available here. Please also see the release announcement in the GAP Forum.

Madeleine Whybrow’s second visit to St Andrews

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.

Madeleine Whybrow

8th International Workshop on Parallel Symbolic Computation (PASCO’17)

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)

Wedderga package coding sprint

In November 2017, Sugandha Maheshwary (IISER Mohali, India) and Ángel del Río (Murcia, Spain) visited St Andrews for a Wedderga package coding sprint. The main goal of the coding sprint was to incorporate new code developed by Sugandha Maheshwary and Gurmeet Bakshi, and prepare a new release of the package which will be compatible with the new major release of GAP coming soon in 2018. The sprint resulted in the new major release of Wedderga. Beyond incorporating the new code, we improved other aspects of the package, including code coverage by the regression tests. In her reflections on the visit, Sugandha Maheshwary wrote:

I have been using and working with GAP for more than 3 years and wanted to contribute some functions. I visited University of St Andrews as a visitor of Dr Alexander Konovalov. During my stay I was trained for technical skills that are needed for the collaborative software development. I was part of the process of release of new version (4.9.0) of package named “Wedderga”, which gave me immense pleasure, confidence and motivation to contribute further. I was also introduced to Software Carpentry, which I would say is a must for beginners. My visit was largely funded by CoDiMa, and I am really grateful for the same.

Carnival of Mathematics #150

Welcome to the 150th Carnival of Mathematics. This is a monthly digest of selected mathematical blogs, hosted each month on a different site. The 1st Carnival has been published in February 2007, so this tradition already continues for more than 10 years. Thanks to everyone who submitted their blog posts to the 150th Carnival!

Following the tradition, we first ask what do we know about 150. It is the number of groups of order 900, the number of integer solutions to x2+y2+z2 = 152 (allowing zeros and distinguishing signs and order), and the number of squares on the 5x5x5 Rubik’s cube. It is a Niven number (aka Harshad number), i.e. it is divisible by the sum of its digits, and an abundant number, i.e. the sum of its divisors exceeds 2×150.

Now to the highlights of some recent mathematical blogs. Ben Orlin, the author of Math with Bad Drawings blog, published new post (with drawings!) called The State of Being Stuck. It is based on Andrew Wiles’ answer to the question by Ben asked at the Heidelberg Laureate Forum 2016. Being stuck is a part of the research process, and is not something to be afraid of – this is what Andrew Wiles would like to emphasize when talking about mathematics to a broader public. This year, Ben went to the Heidelberg Laureate Forum again, and you may find his account of the event here. Another blog post about the Heidelberg Laureate Forum 2017 has been published by Katie Steckles on the Aperiodical.

And if you’re stuck at something and need a break, then perhaps you may find inspiration through looking at nature’s beauty – for example, LThMath suggests to look at Symmetry and Butterflies, which may reflect concepts from various areas of mathematics, ranging from analysis to algebra and statistics.

Speaking of statistics, John D. Cook discusses an interesting application in Randomized response, privacy, and Bayes theorem. Suppose you have a database with sensitive information, and you would like deliberately corrupt it with random noise to anonymise records. How this can be done in a way to preserve privacy while still keeping the data statistically useful?

Rachel Traylor wrote several posts for The Math Citadel website: The Central Limit Theorem Isn’t A Statistical Silver Bullet, where she shows how The Central Limit Theorem, for all its power and popularity, is not a one-stop result to be used for all occasions, and Cauchy Sequences: The Importance Of Getting Close. The latter is a wonderfully accessible explanation of the Cauchy property of sequences, taking time to rigorously examine every piece of the definition.

Anthony Bonato has been interviewing prominent mathematicians in a series of blog posts, most recently Eugenia Cheng, researcher in category theory and the author of How to Bake Pi and Beyond Infinity, and Jennifer Chayes, one of the leading researchers in network science, working at the interface of mathematics, physics, computational science and biology.

Now to discrete mathematics. Reduce The Problem: Permutations And Modulo Arithmetic is another post by Rachel Traylor which in a very accessible manner explains permutation and introduces the concept of isomorphism.

In EKR, Steiner systems, association schemes, and all that, Peter Cameron discussed a class of graphs which give a context to two major results in combinatorial mathematics: the construction of Steiner systems, and the Erdős–Ko–Rado theorem (this story has a continuation in his later post here). Steiner systems also lead to a winning strategy in the card game described in the post MINIMOGs and Mathematical blackjack by David Joyner.

In the Quanta Magazine, Erica Klarreich writes about the recent result on O’Nan moonshine by Ken Ono, John Duncan and Michael Mertens (see their paper Pariah moonshine in Nature Communications) in her article Moonshine Link Discovered for Pariah Symmetries. Another new article in the Quanta Magazine is Mathematicians Measure Infinities and Find They’re Equal by Kevin Hartnett. It introduces recent results by Maryanthe Malliaris and Saharon Shelah, which lead to their Third Hausdorff Medal 2017 award.

Ian Gent published Why the world’s toughest maths problems are much harder than a chess puzzle, and well worth US$1m at The Conversation. There he explains the n-queens completion problem and gives some comments to the recent paper written by Chris Jefferson, Peter Nightingale and Ian Gent and published in the Journal of Artificial Intelligence Research, where they show that this problem is NP-complete.

Several posts in September reported news on mathematical software and its applications, in particular Parallel multivariate multiplication by Bill Hart, and Types of Gaussian Elimination and more technical High Performance Meataxe Interface redesign by Richard Parker. Katie Steckles reported about the new largest generalised Fermat prime, discovered by the PrimeGrid project.

Next Carnival

151th Carnival of Mathematics will be hosted by Frederick at White Group Mathematics. Please see the main Carnival website for further details and the form to submit blog posts to the new Carnival.