Programme

Abstracts for plenary talks

Eamonn O'Brien: "Matrix Group Recognition: status and future?" (slides)

Over the past decades, much progress has been achieved on developing high-quality algorithms to answer questions about matrix groups defined over finite fields. We will briefly summarise this project and identify the individual critical pieces needed to support the necessary framework. We will also mention some hard challenge problems.

A. Borovik and S. Yalcinkaya: "Black box algebra and homomorphic encryption"

The principal requirement for homomorphic encryption is that it should preserve algebraic operations on the set of plaintexts and allow computations with ciphertexts (encrypted data) without access to their actual values. Black box algebra is an umbrella term for probabilistic methods of computing in finite algebraic structures: groups, fields, rings, etc.. widely used in computational algebra. The two fields of research have happened to be closely interconnected, which gives some insights into potential weaknesses of homomorphic encryption schemes.

Alexander Hulpke: "What Can Composition Trees Do For You" (slides)

The output of matrix group recognition (and indeed some of the ingredients of matrix group recognition) is not a goal in itself, but a tool for working with groups. I will describe how this is used already, and could be used further, in GAP. I will also describe what functionality is not available (but would have a high pay-off) and describe a number of problems that will need to be resolved to utilize such a tool effectively.

Cheryl Praeger: "Estimation, Probability bounds and complexity of algorithms" (slides)

Success of a randomised algorithm in computational group theory often depends on finding a small subset of elements that acts as a “witness” for the truth of some assertion. The algorithm performs well if the likelihood of finding a witness subset is high. These rather vague statements need to be made precise to prove that an algorithm is valid, or to bound its error probability, or to determine its complexity. Often there is an underlying estimation problem that needs to be solved: the solution yields probability bounds as well as enabling a complexity analysis. I will illustrate these ideas with reference to some simple randomised algorithms for groups.

Colva Roney-Dougal: "Constructive recognition and generating functions"

One of the key achievements of the Matrix Group Recognition Project was the design of constructive recognition algorithms for quasisimple classical groups. In 2009, Leedham-Green and O’Brien gave an algorithm to construct an isomorphism between a given matrix group and a classical group C in its natural representation. This algorithm has proven to be ‘unreasonably effective’ in practice, running much faster than its complexity analysis implies. To improve the complexity analysis requires a greater understanding of the proportions of certain pairs of well-behaved elements of C. This talk will start with an introduction to constructive recognition in general, before presenting some recent work which completes this analysis for certain C. This work is due to Praeger, Dixon and Seress for C = SL(d, q), and is joint work with Glasby and Praeger for C = SU(d,q).

Abstracts for Sessions

Working sessions

Part of the motivation for this summer school is to improve the state of group recognition within the computer algebra system GAP, in particular in the recog package for GAP. We hope that some of the participants will be interested in assisting with this. To this end, we are describing some possible projects in this direction that could already be started during the conference. Different projects require different kinds of skills. It would be great if people with different skills could team up and collaborate on some of these projects. We prepared a list of possible projects, please click here.

Alice Niemeyer, Rebecca Waldecker: Getting to know each other and coordination of projects

TBA

Max Horn: "Infrastructure of the Recog package" (slides)

This session is intended to be an introduction for anybody who is interested in contributing to the GAP package recog. A rough outline:

• A brief report on the status quo: What is there, and what is missing?
• A brief overview of the technical infrastructure used to develop recog (e.g. GitHub code repository, issue tracker, Travis tests, Code coverage, ...)
• The main part of the session will be a guide to the inner workings of recog, with both a high-level overview, but also an introduction to the technical side (e.g. which function does what? How to add a new recogntion method? ...)
• Segue to a discussion on how to contribute (see also the list of projects)

Frank Lübeck: "Constructing groups and representations in GAP"

TBA

Eamonn O'Brien: "Coordination of further work and open problems"

TBA

Alice Niemeyer, Eamonn O'Brien: "Closing discussion"

TBA