Conferences

'Local Mathematics' at the Literary and Philosophical Societies

New Directions in the History of Mathematics: Research and Teaching, The Open University, 25th April 2024 (invited speaker)

From the end of the eighteenth century, a large number of so-called ‘literary and philosophical societies’ were established, mainly across the north of England. Often associated with a museum or a library, these organisations arranged public lectures in a variety of subjects, usually aimed at newly wealthy members of the middle classes who had not been able to attend university. In their founding statements, these societies tended to be rather ambitious, professing an interest in all possible (scientific) subjects. In practice, however, they almost always became rather narrower in their scope, structured around the subjects that were of interest to their most active members. Thus, despite an initial assertion of an interest in the subject, mathematics rarely featured in their lecture programmes. But it was not absent altogether. In this talk, I will examine the ways in which mathematics or topics related to mathematics were communicated (or not) to non-academic audiences under the auspices of the literary and philosophical societies.

How do we know what we know about ancient Egyptian mathematics?

Invited lecture to 8th British Early Career Mathematicians' Colloquium, University of Birmingham, 21st July 2023

An outline of the basic principles of ancient Egyptian mathematics appears at the beginning of many a modern book on the history of mathematics, but there is often little reflection on where this information actually comes from. The surviving ancient sources that deal with the mathematics of Egypt are surprisingly few. Most of them came to the attention of scholars in the second half of the nineteenth century, and have been studied variously by Egyptologists, mathematicians, and historians of mathematics, each of whom have brought very different attitudes to the subject. In this lecture, my aim will be to attempt to answer the question in the title, and to contrast the different approaches and assumptions that have been brought to this area by scholars from diverse backgrounds.

Integration across borders? The early International Congresses of Mathematicians

Invited lecture to 12th Early Career Researchers in Mathematics Conference, University of Durham, 13th July 2023

The International Congresses of Mathematicians (ICMs) have been held at (reasonably) regular intervals since 1897, and have certainly not been isolated from the wider political circumstances within which they have taken place – despite the apolitical image that their organisers have often tried to present. In this lecture, I will give an overview of the early ICMs, leading up to the particularly interesting example of the 1936 congress, held in Oslo in a climate of growing tension in European politics. I will discuss the motivations of the Norwegian organisers, the activities of the Nazi-led German delegation, and the reasons for the absence of all Italian and Soviet mathematicians.

The Oslo International Congress of Mathematicians 1936

Session on History of Mathematics with a focus on Scandinavian Mathematics, 29th Nordic Congress of Mathematicians, Aalborg, 5th July 2023

The International Congresses of Mathematicians (ICMs) have been held at (reasonably) regular intervals since 1897, and have certainly not been isolated from the wider political circumstances within which they have taken place. This is particularly true of the 1936 ICM, held in Oslo. In this talk, I give a brief account of this congress, commenting on the mathematics presented there, but mainly focusing on the motivations and positioning of the Norwegian organisers, the activities of the Nazi-led German delegation, and the reasons for the Italian and Soviet absences. The talk draws its content from the 2020 book Meeting under the Integral Sign? The Oslo Congress of Mathematicians on the eve of the Second World War by Christopher D. Hollings and Reinhard Siegmund-Schultze, and also draws upon the research of Henrik Kragh Sørensen.

The Oslo International Congress of Mathematicians 1936

History of Mathematics through Collaboration: Toward a Composite Portrait of Oswald Veblen, Mathematisches Forschungsinstitut Oberwolfach, 8th December 2022

The International Congresses of Mathematicians (ICMs) have been held at (reasonably) regular intervals since 1897, and have certainly not been isolated from the wider political circumstances within which they took place. This is particularly true of the 1936 ICM, held in Oslo. In this talk, I give a brief account of this congress, commenting on the mathematics presented there, but mainly focusing on the activities of the Nazi-led German delegation, and the reasons for the Italian and Soviet absences. Throughout, I will try to bring out Veblen’s contributions both to the ICMs (he was a plenary speaker in Oslo) and to the wider international mathematical community.

Triangulating ancient Egyptian mathematics

BSHM Christmas Meeting (online), 3rd December 2022

When the details of ancient Egyptian mathematics were being reassembled in the late-nineteenth and early-twentieth centuries, scholars were able to draw upon two distinct types of information within the surviving sources. In the first instance, the growing understanding of ancient Egyptian languages and scripts made it possible for some basic meaning to be extracted from the texts. Reconstruction of the ancient languages was, however, an on-going process, and so wherever readings were uncertain, it was possible, and necessary, to interpret texts in the light of an understanding of how the mathematical content ought (from a modern point of view) to work. For the most part, these two sources of evidence, the philological and the mathematical, complemented each other. In rare instances, however, they appeared to clash. In this talk, I will examine one such instance, that of Problem 51 of the Rhind Mathematical Papyrus, concerning the area of a triangle, in which philological and mathematical evidence seemed to point in different directions.

George Leigh Cooke and Bartholomew Price

Oxford's Sedleian Professors of Natural Philosophy: The First 400 Years, Weston Library, Oxford, 18th June 2022

A short survey of the lives and work of the two nineteenth-century holders of Oxford's Sedleian Chair in Natural Philosophy: George Leigh Cooke (1779–1853) and Bartholomew Price (1818–1898).

How do we know what we know about ancient Egyptian mathematics?

BSHM Meeting in Memory of Peter M. Neumann, The Queen's College, Oxford, 9th April 2022

I will speak about a project that was strongly encouraged by Peter, and which began with a chance find in a library in Oxford. After briefly establishing some background details, I will give a whistle-stop answer to the question posed in the title, beginning in the ancient world and ending in the early twentieth century. A central theme of the later parts of the story will be the comparison of how Egyptologists and mathematicians have approached the subject of ancient mathematics, and what they have sought from it.

Diversifying the Curriculum in Oxford

Mathematics: Inclusive or Exclusive? Putting colour, culture and context into the curriculum, International Centre for Mathematical Sciences, Edinburgh (online), 25th January 2022

A very brief account of three initiatives that I've been involved in to try to present a more diverse image of mathematics within the Oxford teaching setting: the Diversifying STEM Curriculum Project, a tutorial exercise in group theory involving Australian Aboriginal kinship relations, and some posters covering 'A World History of Mathematics'.

Editing the Rhind Mathematical Papyrus

Symposium 'The shaping of differences in the historiography of ancient mathematics', 26th International Congress of History of Science and Technology, Prague (online), 30th July 2021

The Rhind Mathematical Papyrus (RMP) was found in Luxor in 1858, and purchased by the British Museum six years later. It dates from around 1500 BCE and consists of over 80 arithmetical and geometrical problems of types that would probably have been encountered during a scribal career. The discovery of the RMP revolutionised modern knowledge of ancient Egyptian mathematics, which up to this point had been understood only in the vaguest of terms. A comprehensive study of the contents of the papyrus was published by the Egyptologist August Eisenlohr in the 1870s, and for many years any discussion of the RMP simply repeated his findings. By the end of the nineteenth century, however, developments in the understanding of the Egyptian language and hieratic script meant that a new edition was called for; this finally appeared in 1923, edited by the (mathematically trained) Egyptologist T. Eric Peet, who was able to provide a more fully contextualised interpretation of the RMP. A further edition of the papyrus appeared in print at the end of the 1920s, edited by A. B. Chace, and aimed very much more at mathematicians than at Egyptologists. Indeed, in the decades that followed, Chace's edition became the go-to version of the RMP for mathematicians, whilst Peet’s was usually favoured by Egyptologists. Starting from comments made by Peet himself, this talk will consider the different approaches of mathematicians and Egyptologists to the RMP, and describe how these have shaped the study of ancient Egyptian mathematics more generally.

Thomas Eric Peet, historian of mathematics

People, Places, Practices: Joint BSHM-CSHPM/SCHPM conference, University of St Andrews (online), 12th July 2021

In 1923, the English Egyptologist Thomas Eric Peet (1882–1934) published an edition of the Rhind Mathematical Papyrus – one of the main sources, held in the British Museum (P. BM EA 10057-8), on ancient Egyptian mathematics. Although a facsimile of the papyrus had been published at the end of the nineteenth century, and certain aspects of it had been studied by other Egyptologists, Peet's edition, translation and study was the first comprehensive treatment of its mathematical content. In his commentary on the papyrus, as well as in the small number of other works that he published on ancient Egyptian mathematics, Peet displayed a sensitivity to historical context that was not present in the works of most other historians of ancient mathematics during the 1920s and 1930s. Perhaps for this reason, Peet's work on ancient mathematics (his edition of the Rhind papyrus aside) appears to be little known beyond Egyptological circles. In this talk, I will describe the content of Peet's study of ancient Egyptian mathematics, and consider his approach to the subject in comparison to that of contemporaneous historians of mathematics.

Reading the Rhind Mathematical Papyrus

Maynooth Conference in the History of Mathematics, 2nd August 2019

The Rhind Mathematical Papyrus is one of the major sources available to us on ancient Egyptian mathematics: it consists of 87 arithmetical and geometrical problems, grouped by theme; the papyrus is a compendium of different types of problems that would be encountered during a scribal career. Some parts of the papyrus were published in the later decades of the nineteenth century, but it was not until 1923 that a comprehensive edition finally appeared in print: that of Thomas Eric Peet (1882–1934), then Professor of Egyptology in Liverpool. Peet’s edition remains the standard version of the text for Egyptologists, although a 1927 edition by Arnold Buffum Chace (1845–1932) has often been preferred by mathematicians. In this talk, I will consider the different requirements of these two groups of readers, and investigate how they each engaged with the editions of the papyrus. This study draws heavily upon material available in the Griffith Archive in Oxford, as well as a number of annotated copies of Peet’s edition of the Rhind papyrus that are available in Oxford libraries; in these copies, we find marginal annotations of two clear types: those made by Egyptologists, and those of mathematicians.

Abstraction and axioms: some parallels between 19th-century British and 20th-century American mathematics

AMS Special Session on the History of Mathematics, Joint Mathematics Meetings (AMS/MAA), San Diego, 11th January 2018

During the nineteenth century, we see early examples of both abstract and axiomatic approaches to algebra in the works of several British mathematicians, most notably Augustus De Morgan. We also see criticisms of the associated methods. In the early decades of the twentieth century, similar ideas took a prominent place in the American mathematical community, though apparently largely independently of the prior British work. In this talk, I will look at the similarities that are present in the algebraic works of these two communities, and compare the points upon which each was criticised.

A privileged position? The place of mathematics in Cold War scientific exchange

Genealogies of Knowledge I: Translating Political and Scientific Thought across Time and Space, Manchester, 7th December 2017

During the years of the Cold War, concerns were often raised in the West regarding the apparent ease with which Soviet scientists were able to access Western scientific developments, in contrast to the difficulties often experienced in the opposite direction. Although physical access to published materials was often problematic, the key issue here is language: the dominant Western scientific languages of French, German and English usually posed little problem for Soviet readers, whilst Westerners had considerably greater difficulty in understanding Russian. The launch of Sputnik I in 1957 resulted in a greater Western (particularly American) sense of urgency in engaging with the Russian language. Efforts to teach Russian to scientists met with only limited success, however, and so systematic scientific translations from Russian gradually became the principal means by which Western scientists were able to learn about Soviet research. Opinions as to the value of such translations varied from discipline to discipline, but one area that saw particularly extensive translation efforts was mathematics. These efforts were driven in large part by a high regard in the West for Soviet mathematics, and may also owe their success to the nature of mathematical Russian: that, perhaps more so than in most other disciplines, a knowledge of mathematics enables the reader to glean some small amount of understanding, even if they know no Russian. In this talk, I will argue that for this reason mathematics enjoyed a privileged position in the communication of scientific ideas across Cold War language barriers, which may have served to emphasise the value of scientific translations more generally, but at the same time had a significant effect on the language skills of mathematicians in both East and West.

The postulate analysis of groups in the early twentieth century

4th International Conference on History and Pedagogy of Modern Mathematics, Chengdu, China, 22nd August 2017

In the early years of the twentieth century, so-called 'postulate analysis' – the study of systems of axioms for mathematical objects for their own sake – was regarded by some as a vital part of the efforts to understand those objects. In this talk, I consider the place of postulate analysis within early twentieth-century mathematics by focusing on the example of a group: I outline the axiomatic studies to which groups were subjected at this time, and consider the changing attitudes towards such investigations.

A Language of its Own? Communicating Mathematics across Language Barriers

Symposium on 'Global Mathematics', 25th International Congress of History of Science and Technology, Rio de Janeiro, 26th July 2017

During the years of the Cold War, Western mathematicians developed a great interest in Soviet mathematics, which they perceived as world-leading, and therefore engaged in many efforts to gain greater access to the relevant work. On the language side, these ranged from the teaching of courses in mathematical Russian to the cover-to-cover translation of major Soviet journals. Indeed, where Western resources for accessing Russian-language materials were concerned, mathematics was one of the best-served disciplines. I believe that this enthusiasm was due, at least in part, to the nature of mathematical Russian: that a knowledge of mathematics enables the reader to glean some small amount of understanding, even if they know no Russian. In this talk, I will extend this suggestion to other languages, and argue that the near-universal nature of mathematical notation, terminology and writing style (certainly in recent centuries) has given mathematics a privileged position in the communication of scientific ideas across language barriers.

Meeting under the integral sign? Strategies Shaping the International Congress of Mathematicians, Oslo 1936

BSHS Annual Conference, University of York, 9th July 2017

Mathematics at the Philosophical and Literary Societies

Beyond the Academy. The Practice of Mathematics from the Renaissance to the Nineteenth Century, York, 7th April 2017

The founders of the Philosophical and Literary Societies that emerged across Britain in the late-eighteenth and early-nineteenth centuries typically professed an interest in all strands of knowledge, including mathematics. In practice, however, the lecture programme of a typical Society rarely featured topics that might permit any mathematical content — although there are notable exceptions: the Manchester Literary and Philosophical Society, for instance. In this short talk, I will examine the handling of mathematics by a number of such Societies, and compare this with the central position occupied by mathematics at meetings of the British Association for the Advancement of Science.

A Tour of Group Generalisations of the 1920s and 1930s

Special Session on the History of Mathematics, Joint Mathematics Meeting (AMS/MAA), Atlanta, 5th January 2017

The process of generalisation in mathematics sometimes gets a bad press owing to the presence in the literature of many an ill-motivated 'generalisation for generalisation's sake'. However, a number of different generalisations of the group concept emerged in the 1920s and 1930s, each designed to solve a particular problem. Although not all of these newly defined objects went on to receive broader study, they are nevertheless good examples of well-motivated generalisation. I will give a survey of a selection of these, and point out some of their interesting interconnections.

Reading between the lines: Soviet mathematical biography

Mathematical Biography: a MacTutor Celebration, St Andrews, 17th September 2016

Like any nation, the USSR sought to celebrate its scientific figures, with mathematicians receiving a great deal of attention: obituaries and celebrations of prominent mathematicians featured frequently in leading mathematical journals. However, these biographies are of a very particular style: they are quite impersonal, and could almost have been written as a series of bullet points. Moreover, they are rather selective in the details that they present (true of any biography?): features of the subject's life that did not sit well with the official Soviet line were omitted, whilst other more acceptable points (that would later be quietly dropped from post-Soviet biographies of the same subject) were given a prominent place. In this talk, I will note the perils and pitfalls of working with such biographies, and discuss what we can learn from what they don't tell us.

"Dispelling ignorance and overcoming prejudice": breaching the language barrier in Cold War mathematics

Mathematical Communication during the Cold War, Mathematical Institute, Oxford, 8th July 2016

I discuss the nature of the language barrier in Cold War mathematical communication, and assess the abilities and efforts of various parties to overcome it: for example, the use of Western European languages in Soviet papers, the publication of a journal entirely in French, German and English in what was then Czechoslovakia, and the attempts to educate Western native-English-speaking mathematicians in the use of other languages. In the final part of the talk, I focus specifically on English-speaking Westerners and assess the effectiveness of scientific foreign-language teaching and the impact of the systematic translation of Soviet resources.

R. S. Ball's 'Mechanics' and the 'London Science Class-Books', or Bringing Newton to the masses?

Session on 'Mathematics and Society', 58th British Applied Mathematics Colloquium, Mathematical Institute, Oxford, 6th April 2016

The 'London Science Class-books' were a series of short textbooks on elementary scientific topics that were published in the late 19th century. Intended for use in school teaching, their purpose was not merely to present a sequence of facts, but to explain "as fully as possible the nature of the methods of inquiry and reasoning" by which these facts were obtained. In this talk, I will consider the series' handling of mathematics, with a particular focus on the text on mechanics, authored by the then-Royal Astronomer for Ireland, Robert Stawell Ball.

Russian participation in the early International Congresses of Mathematicians

Special Session on the History of Mathematics, Joint Mathematics Meeting (AMS/MAA), Washington State Convention Center, Seattle, 8th January 2016

The picture of Russian/Soviet attendance at the International Congresses of Mathematicians is a very varied one, with strong delegations at some congresses (particularly the ones of the later Soviet era) and conspicuous absences from others (those of 1936 and 1950 most especially). In this talk, I will examine Russian attendance of the early ICMs, namely, those that took place before the October Revolution. We will see that in this period, many Russian mathematicians were active and enthusiastic participants at such international events.

The mathematical correspondence of Ada Lovelace and Augustus De Morgan

Ada Lovelace Symposium, Mathematical Institute, Oxford, 10th December 2015

During the years 1840-1, Ada Lovelace corresponded with the prominent mathematician Augustus De Morgan, who tutored her in a range of mathematical subjects, including algebra, trigonometry and elementary calculus. Previous readings of this correspondence have resulted in wildly differing assessments of her mathematical abilities, but without any in-depth analysis of the mathematics. In this talk, I will report on a recent new study of the mathematics Ada Lovelace was learning with De Morgan. I will provide what I hope will at last be an accurate unbiased evaluation of her mathematical proficiency.

Soviet views of early (English) algebra

Joint meeting of the BSHM and CSHPM within MAA MathFest, Washington, DC, 6th August 2015

The history of mathematics emerged as a significant discipline in the USSR during the 1930s, apparently building on an earlier Russian interest. In its early stages, it was marked by two major characteristics: a nationalist tenor, and a concern over ideology. The former led to a focus on the contributions of Russian mathematicians, whilst the latter, occasionally at odds with the former, sought to reinterpret the works of historical Russian mathematicians in terms of Soviet ideology. However, as the Soviet study of the history of mathematics opened up after Stalin's death, we find the names of other (non-Russian) historical mathematicians beginning to appear as the subjects of published works. In this talk, I examine the treatment of early algebraists (particularly those in England) at the hands of Soviet authors.

"A batch of observations & enquiries": the correspondence of Ada Lovelace and Augustus De Morgan

LMS–BSHM 150th Anniversary De Morgan Day, De Morgan House, London, 9th May 2015

During the years 1840–1841, Augustus De Morgan corresponded with Ada King, Countess of Lovelace, on mathematical subjects, tutoring her in elementary calculus, amongst other topics. Previous readings of this correspondence have resulted in wildly differing assessments of Lovelace's mathematical abilities, though seemingly without any in-depth analysis of the mathematics. In this talk, I will report on work that is underway to determine what mathematics Lovelace was learning with De Morgan, and finally, it is hoped, to provide an accurate, unbiased evaluation of her mathematical proficiency.

Language use in Soviet mathematics journals

History of mathematics mini-symposium, 4th Joint BMC/BAMC, University of Cambridge, 31st March 2015

The (linguistic) problems experienced by Western mathematicians in their attempts to access the mathematical work of the Soviet Union during the years of the Cold War are well documented. I will begin with a short discussion of these, before moving onto the rather easier situation in earlier decades: during the 1920s and 1930s, several Soviet journals employed Western languages in order to reach a wider international audience. I provide a brief analysis of the languages used, and discuss the reasons for the switch, following the Second World War, to the exclusive use of Russian.

Soviet participation in the ICMs: the early stages of a study

Mathematics: place, production and publication, 1730-1940, Institut Mittag-Leffler, Stockholm, 26th January 2015

A short account of work in progress.

Mathematics across the Iron Curtain

BSHM Christmas Meeting, The Birmingham and Midland Institute, Birmingham, 7th December 2013

I give a brief account of the communications difficulties experienced by Cold War mathematicians in their efforts to learn more about the work of their counterparts on the opposite side of the Iron Curtain. The focus will be largely (though not exclusively) on the experiences of Soviet mathematicians. I deal first with the problems afflicting personal contacts (correspondence and conference attendance), and then, more briefly, with the difficulties in accessing the publications of 'the other side'.

An Obsession for Documentation: Surveys of Mathematical Progress in the USSR

History of mathematics workshop, 65th British Mathematical Colloquium, University of Sheffield, 27th March 2013

Mathematics was one of the most successful sciences to be pursued in the USSR, with many Soviet mathematicians achieving worldwide fame. Perhaps as a documentary basis for international boasting, a number of official surveys were commissioned on the progress of Soviet mathematics. These appeared at intervals, and thereby give us a series of snapshots of Soviet mathematics down the decades. I will give an overview of the surveys that are available, and indicate what they can tell us about the study of mathematics in the USSR.

A. H. Clifford and Unique Factorisation: Some Forgotten Early Steps into Semigroup Theory

Groups and Semigroups: Interactions and Computations, Faculdade das Ciências, Universidade de Lisboa, 26th July 2011

In this historical talk, I will discuss the notion of prime factorisation in semigroups, with a special emphasis on the work of A. H. Clifford in the 1930s. Clifford's work was inspired by that of Emmy Noether in the ring case, but his approach also owes much to the work of his doctoral supervisors, E. T. Bell and Morgan Ward, and their attempts to provide a sound, postulational basis for arithmetic. I will describe the origins of Clifford's work, which was not only Clifford's first work in semigroup theory, but was also amongst the first semigroup theory of any kind.

An ESN Theorem for ∧-premorphisms

Workshop on Groups, Semigroups and Applications, Centro de Álgebra da Universidade de Lisboa, 24th April 2009

The Ehresmann-Schein-Nambooripad (ESN) Theorem states that the category of inverse semigroups and morphisms is isomorphic to the category of inductive groupoids and inductive functors, and that the category of inverse semigroups and ∨-premorphisms is isomorphic to the category of inductive groupoids and ordered functors, where a '∨-premorphism' is a function between inverse semigroups introduced by McAlister and Reilly in 1977. McAlister and Reilly also defined the dual notion of a '∧-premorphism' and this type of premorphism has found an application in recent years in the study of partial actions: loosely speaking, a partial action is equivalent to a premorphism in much the same way that an action is equivalent to a morphism. In this talk, I will discuss an ESN-type theorem for inverse semigroups and ordered ∧-premorphisms.

Partial Actions of Inverse Monoids on K-Rings

Mini-workshop on algebra, Centro de Álgebra da Universidade de Lisboa, 18th April 2008

The partial actions of groups on K-rings (a.k.a. associative K-algebras) have been studied by Dokuchaev and Exel (2005), as a purely algebraic version of earlier work on the partial actions of groups on C*-algebras. In particular, Dokuchaev and Exel address the perennial problem of constructing an action from a partial action, which in this case is termed the 'enveloping action' of the given partial action. In this talk, I will set up appropriate definitions for the partial actions of inverse monoids on K-rings and describe the construction of enveloping actions for such partial actions.

Partial Actions of Inverse Monoids on K-Rings

Semigroup splinter group, 60th British Mathematical Colloquium, University of York, 26th March 2008

Constructing Incomplete Actions

Conference on Semigroup, Acts and Categories, with Applications to Graphs, Tartu Ülikool, Estonia, 29th June 2007

The partial actions of monoids on sets have been studied as a natural generalisation of the partial group actions of Kellendonk and Lawson. In particular, the question of whether a (global) action may be constructed from a partial action has been investigated. One such method for achieving this is that by globalisation, whereby the acting group/monoid is fixed, whilst the set being acted upon is enlarged in such a way that the original group/monoid acts (globally) on this new set. Globalisation results have been found both in the group case, and for arbitrary monoids.

Of particular interest is the study of the partial actions of a class of monoids called weakly left E-ample monoids. These monoids generalise inverse monoids and arise very naturally as (2,1,0)-subalgebras of partial transformation monoids. We find, however, that if we are to 'globalise' the partial action of a weakly left E-ample monoid in such a way that its unary operation is respected, then we cannot construct a global action, as before, but we must instead settle for a slightly weaker notion of 'action' which we term an incomplete action. In this talk, I will discuss the definition of the partial action of a weakly left E-ample semigroup on a set, before presenting the results on the construction of the corresponding incomplete action. I will point out how these results may then be specialised to the inverse case.

Partial Actions of Monoids

Fountainfest, University of York, 14th October 2006

I will discuss partial monoid actions, in the sense of Megrelishvili & Schröder (2004). These are equivalent to a class of premorphisms, called strong premorphisms. There are two distinct methods for constructing a monoid action from a partial monoid action. I will describe one of these: the expansion method, which leads to a generalisation of a result of Kellendonk & Lawson (2004) from the group case.