Seminars

Side-stepping absurdity: Some strange ideas about algebra in 19th-century Britain

Talk to the Oxford Invariants, Mathematical Institute, Oxford, 23rd November 2021

By the end of the 18th century, mathematicians had developed a range of powerful techniques for manipulating and solving polynomial equations. Many new ideas, such as complex numbers, had emerged along the way, and these had been accepted (mostly) without trouble into the mathematician’s day-to-day toolkit. In 1790s, however, some mathematicians in Britain began to question the validity not only of complex numbers but even of negative numbers, and proposed to rebuild algebra in such a way that negative numbers could never appear. Although few practicing mathematicians ever seriously considered rejecting negative numbers in this way, the debate did raise questions about the way in which algebra should be approached - in particular, how it should be taught. In this talk, I will look at this episode in the history of algebra and describe where it fits into the mathematical landscape of 19th-century Britain.

How international were the early International Congresses of Mathematicians?

History of Mathematics in India Project (HOMI), Seminar Series ‘History of Mathematics’, Indian Institute of Technology, Gandhinagar, delivered online, 18th June 2021

The International Congresses of Mathematicians (ICMs) were created at the end of the nineteenth century as conferences at which the mathematicians of many countries could come together. Although they are now held all around the world and have a claim to being truly international, the earlier meetings were attended almost exclusively by mathematicians from Europe and North America. In this talk, I will survey the early history of the ICMs, and discuss how they gradually opened up to the wider world.

Meeting under the integral sign? The Oslo Congress of Mathematicians on the eve of the Second World War

Séminaire 'Histoire et philosophie des mathématiques, 19è–21è siècles', Université de Paris, delivered online, 16th April 2021

Oral communication, and congresses in particular, remain a crucial element within mathematical communication – even in the current age of electronic mail. Indeed, congresses and meetings serve many more purposes than simply communicating information about recent mathematical research, and this has always been the case, with each particular historical period setting different priorities. The present talk is drawn from a book, recently completed with Reinhard Siegmund-Schultze, which focuses upon and stresses the historically unique character of the Oslo International Congress of Mathematicians of 1936. This congress was the only one on this level to be held during the period of the Nazi regime in Germany (1933–1945) and after the wave of emigrations from it. In this talk, I will survey the differences between the goals of the various participants in the congress, most particularly the Norwegian organisers, and the Nazi-led German delegation. I will consider also the background to the absence of the proposed Soviet and Italian delegations. If time permits, I will also go into the mathematical dimension of the Oslo congress, this being the conference at which the Fields Medals were awarded for the first time. Overall, the Oslo congress may be used as a lens through which to view the state of the art of mathematics in the mid-1930s.

Mathematics, from Ancient Egypt to Modern Academia (with R. B. Parkinson)

Talk to the Oxford Invariants, delivered online, 9th November 2020

The subject of ancient Egyptian mathematics typically features, alongside its Babylonian counterpart, at the beginning of almost any general history of mathematics. But how do we know what Egyptian mathematics looked like? The understanding of ancient Egyptian arithmetic and geometry emerged slowly during the nineteenth and early twentieth centuries, thanks to the work both of Egyptologists and of mathematicians. However, the two groups often had different interpretations of the surviving evidence. In this talk, we'll describe how ancient Egyptian arithmetic worked, and use this to contrast these different approaches to the subject.

The 1936 Oslo International Congress of Mathematicians

Oxford History of Mathematics Forum, delivered online, 25th May 2020

Overview of the content of the book Meeting under the Integral Sign? The Oslo Congress of Mathematicians on the Eve of the Second World War.

The history and historiography of ancient Egyptian mathematics in Oxford

Historisches Seminar: Wissenschaftgeschichte, Goethe-Universität Frankfurt, 10th December 2019

I will describe some ongoing work that Richard Parkinson and I are carrying out concerning the twentieth-century historiography of ancient Egyptian mathematics, with a focus on the role played by Oxford-based figures. The project draws heavily upon some unexpected archival finds, as well as the marginalia in several books held by Oxford libraries.

The history and historiography of ancient Egyptian mathematics in Oxford

Seminar in Egyptology and Ancient Near Eastern Studies, Oriental Institute, University of Oxford, 22nd October 2019

I will describe some ongoing work that Richard Parkinson and I are carrying out concerning the twentieth-century historiography of ancient Egyptian mathematics, with a focus on the role played by Oxford-based figures. The project draws heavily upon some unexpected archival finds, as well as the marginalia in several books held by Oxford libraries.

Teaching approaches to ancient mathematics

Showcasing the Ashmolean Faculty Fellowships, Ashmolean Museum, Oxford, 18th June 2019

A short summary of work undertaken in the museum during Hilary Term 2019 as part of an Ashmolean Faculty Fellowship, funded by the Andrew W. Mellon Foundation.

Language use in Soviet mathematics journals

Séminaire Cirmath: La circulation mathématique en Europe centrale et en Russie (Tchéquie, Russie et Union soviétique), Institut Henri Poincaré, Paris, 27th May 2019

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. Nevertheless, there have been times over the past three centuries when Russian mathematicians have deliberately chosen to publish in Western European languages in order to increase their international readership. I will begin by surveying the mathematical journals that were available in/from Russia prior to the twentieth century, before considering the language policies of journals under the Soviet regime.

Adventures in the history and historiography of ancient Egyptian mathematics

Oxford History of Mathematics Forum, The Queen's College, Oxford, 29th April 2019

I will describe some ongoing work that I have recently been drawn into thanks to a number of unexpected archival finds. This work concerns the ways in which ancient Egyptian mathematics was studied during the twentieth century, and looks at the involvement of some Oxford-based figures.

Wagner's Theory of Generalised Heaps

York Semigroup, Department of Mathematics, University of York, 17th April 2019

I will give a historical and mathematical overview of the notions of heaps, generalised heaps, and semiheaps, as developed by the Russian mathematician V. V. Wagner in the 1950s. These ideas provide an elegant model for the study of systems of binary relations between sets. I will describe the background to Wagner's ideas, and point to their connections with the theory of inverse semigroups.

Meeting under the integral sign? The 1936 Oslo International Congress of Mathematicians

A History of Mathematics Summer Miscellany (History of Mathematics Seminar), Mathematical Institute, Oxford, 27th July 2018

The International Congresses of Mathematicians (ICMs) have taken place at (reasonably) regular intervals since 1897, and although their participants may have wanted to confine these events purely to mathematics, they could not help but be affected by wider world events. This is particularly true of the 1936 ICM, held in Oslo. In this talk, I will give a whistle-stop tour of the early ICMs, before discussing the circumstances of the Oslo meeting, with a particular focus on the activities of the Nazi-led German delegation.

"Black strokes upon white paper": changing attitudes towards symbolic algebra from the 19th into the 20th century

Seminar in the History of the Exact Sciences, All Souls College, Oxford, 17th January 2018

During the first half of the nineteenth century, a debate took place amongst British mathematicians concerning the nature of the symbols used in algebra: did they necessarily stand for numbers, or could they simply be manipulated according to specified rules, with interpretation (if any) coming later? Critics of the former point of view decried the restriction that would thereby be placed upon the use of algebra, whilst those of the latter saw it as being ill-justified and often too far removed from concrete examples. For a range of reasons, both educational and philosophical, a fully abstract 'symbolical algebra' never appeared in nineteenth-century British mathematics; 'abstract algebra' as we now know it derives from largely German sources at the end of the century. Nevertheless, as the abstract point of view came gradually to dominate algebra during the early decades of the twentieth century, similar debates took place to those of a century earlier. This time, however, the abstract approach was received more sympathetically. In this talk, I will contrast these changing attitudes towards abstract/symbolic algebra, and address the question of why this approach became more acceptable in the twentieth century.

What is algebra?

What do historians of mathematics do? Mathematical Institute, Oxford, 8th May 2017

I will address the question of "What do historians of mathematics do?" by turning to another: "What is algebra?" In answering this second question, and surveying the way that the answer changes as we move through the centuries, I will highlight some of the problems that face historians of mathematics when it comes to interpreting historical mathematics, and give a flavour of what it means to study the history of mathematics.

The algebraic theory of semigroups: interactions between US and European mathematics during the 20th century

Séminaire d'Histoire des Mathématiques de l'Institut Henri Poincaré, Paris, 16th December 2016

Some early glimpses of the algebraic theory of semigroups may be seen in the work of both European and American mathematicians in the first decade of the twentieth century, although the theory proper did not begin to develop until the 1920s. Thereafter, contributions to the theory continued to be made on both sides of the Atlantic. These contributions were, however, closely interrelated. In this talk, I will use the theory of semigroups as a lens through which to view the interactions between US and European mathematics during the 20th century.

"Dispelling ignorance and overcoming prejudice": communications difficulties in Cold War science

Seminar in Mathematics and Mathematics Education, Universitetet i Agder, Kristiansand, Norway, 19th September 2016

In this talk, I will give a general overview of the problems associated with East-West scientific communication during the twentieth century, and during the Cold War in particular, with a view to describing how these changed and were overcome over the decades. I will focus in particular on the language barrier in Cold War scientific communication, and assess the abilities and efforts of various parties to breach it. In the final part of the talk, I will 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.

Meeting under the integral sign? The 1936 Oslo International Congress of Mathematicians

Queen's College Symposium, Oxford, 7th June 2016

The International Congresses of Mathematicians (ICMs) have taken place at (reasonably) regular intervals since 1897, and although their participants may have wanted to confine these events purely to mathematics, they could not help but be affected by wider world events. This is particularly true of the 1936 ICM, held in Oslo. In this talk, I will give a whistle-stop tour of the early ICMs, before discussing the circumstances of the Oslo meeting, with a particular focus on the activities of the Nazi-led German delegation.

A brief history of semigroup representations

North British Semigroups and Applications Network, University of York, 11th May 2016

In this talk, I will present an overview of the history of the earlier parts of the study of semigroup representations, beginning with the initial work of Sushkevich and Clifford on semigroups of matrices, before moving on to the parallel developments found in the work of Munn and Ponizovskii.

Russian participation in the early International Congresses of Mathematicians

Oxford History of Mathematics Forum, The Queen's College, 25th 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.

Russian views of English algebra: a preliminary study

Oxford History of Mathematics Forum, The Queen's College, Oxford, 26th May 2014

I give a preliminary account of a tentative project to study the representations of early English algebraists (and mathematicians more generally) in Russian history of mathematics materials, principally from the Soviet era.

Mathematics across the Iron Curtain

Oxford History of Mathematics Forum, The Queen's College, Oxford, 14th October 2013

I give an overview of the communications difficulties experienced by Cold War mathematicians (indeed, scientists more generally) 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 scientists. 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'.

Densely Embedded Ideals: A Handy Tool from the Pages of Soviet Semigroup Theory

York Semigroup, Department of Mathematics, University of York, 10th December 2012

In this historical talk, I will introduce and discuss a notion that was used extensively in the work of Soviet semigroup theorists in the 1950s: that of a densely embedded ideal. Despite the many nice theorems that may be stated in terms of such ideals, they do not seem to have been picked up by Western semigroup theorists. This is something that I hope to address in this talk. After a brief account of the early development of semigroup theory in the USSR, I will introduce the notion of a densely embedded ideal and present some of the pleasing results that they may be used to obtain.

Unique Factorisation from Rings to Semigroups

Algebra and geometry seminar, School of Mathematics, University of Bristol, 5th December 2012

In this historical talk, I will discuss the notion of prime factorisation in semigroups, which was studied extensively in the 1930s by analogy with the corresponding concept for rings. In particular, I will describe the work of A. H. Clifford, which was inspired by that of Emmy Noether in the ring case. Nevertheless, Clifford's approach also owes much to the work of his doctoral supervisors, E. T. Bell and Morgan Ward, and their attempts to provide a postulational basis for arithmetic. I will begin with a brief survey of the study of problems of unique factorisation down the centuries, before describing the origins of Clifford's work, and its role in the birth of semigroup theory.

Adventures in Kharkov

Oxford History of Mathematics Forum, The Queen's College, Oxford, 28th May 2012

I will give a biographical sketch of the Russian algebraist A. K. Sushkevich (1889-1961). In particular, I will survey some of the sources available to me concerning Sushkevich, including some obtained on a research trip to Kharkov (Ukraine), of which I will give a brief account.

Левые рестриктивные полугруппы (Left restriction semigroups)

66th Herzen seminar (algebra section), Herzen Russian State Pedagogical University, St Petersburg, 20th April 2012

Я буду говорить о левых рестриктивных полугрупп и об их связи с левыми индуктивными созвездиями.

I will talk about left restriction semigroups and their connection with left inductive constellations.

The Perils of Taking Shortcuts: Embedding Semigroups in the 1930s

NBSAN, School of Mathematics, University of Manchester, 30th August 2011

I will discuss the problem of embedding (cancellative) semigroups in groups. This was a problem which was first considered in the 1930s, as an analogue of that of embedding rings in fields. Perhaps the most significant contribution in this area was that of A. I. Maltsev in 1939. I will give a brief discussion of Maltsev's work, after having built up to it via the (not always correct) contributions of earlier researchers.

A Short History of Inverse Semigroups

York Semigroup, Department of Mathematics, University of York, 16th February 2011

I will trace the development of the concept of an inverse semigroup from the 'pseudogroups' of Veblen and Whitehead through to V. V. Wagner's 'generalised groups' and their independent discovery by G. B. Preston. Along the way, I will describe the conceptual difficulties which initially hindered the leap from pseudogroups to inverse semigroups.

Beyond the Ehresmann-Schein-Nambooripad Theorem

York Semigroup, Department of Mathematics, University of York, 25th May 2010

The celebrated Ehresmann-Schein-Nambooripad Theorem expresses the deep underlying connection between inverse semigroups and inductive groupoids. The usual statement of the theorem involves the category of inverse semigroups and morphisms, and the category of inverse semigroups and ∨-premorphisms. The functions dual to ∨-premorphisms, ∧-premorphisms, have gained prominence in recent years through their use in the theory of partial actions of semigroups. We therefore describe the derivation of a version of the Ehresmann-Schein-Nambooripad Theorem involving the category of inverse semigroups and ∧-premorphisms.

The Early Days of Semigroup Theory, or Learning to be an Historian

Oxford History of Mathematics Forum, The Queen's College, Oxford, 26th April 2010

I will give an overview of the development of the algebraic theory of semigroups, discussing the work that I have carried out so far, as well as indicating where I hope to go with it in the future. One name which will crop up during this overview will be that of the Russian mathematician Anton Kazimirovich Suschkewitsch, in whose life and work I am particularly interested. In the final part of the talk, I will discuss my efforts to learn more about Suschkewitsch.

Enveloping Actions for Partial Actions on Rings

Seminário de álgebra, Centro de Álgebra da Universidade de Lisboa, 23rd October 2009

A partial action is, loosely speaking, an action which is not everywhere-defined. Partial actions on sets have seen widespread study - first the partial actions of groups on sets, then those of monoids, inverse semigroups, inductive groupoids, etc. In the case of partial actions of groups on sets, for example, it is possible to show that any partial action may be 'completed', to give a full action, termed the 'globalisation'.

Also appearing in the literature are the partial actions of groups on rings and (associative linear) algebras. Given such a partial action, people have once again posed the perennial question concerning partial actions: when/how can we construct a full action from a given partial action? The major construction which has emerged in this context is that of the so-called 'enveloping action'. This is an analogue of globalisation for partial actions on rings. However, in contrast to the situation with the globalisation, not every such partial action admits an enveloping action.

In this seminar, I will begin by surveying the results concerning the construction of an enveloping action for partial actions of groups on rings. I will then move on to consider generalisations of these results to the monoid case, pointing out certain issues which arise, and presenting two cases where these generalisations work particularly well (though slightly differently): those of inverse monoids and right groups.

Partial Group Actions and Their Generalisations

Seminário de álgebra, Centro de Álgebra da Universidade de Lisboa, 30th November 2007

The notion of the partial action of a group on a set first arose in the context of operator algebras and has since found a use in a wide range of other areas: model theory and tilings, for example. One question which has received particular attention is the following: given a partial group action, (when) can one construct an action? In this seminar, I will begin by discussing the definition of a partial group action (with examples), before describing one such method for constructing an action from a partial action: the process of 'globalisation'. This is a construction which appears in a number of other contexts (topology, combinatorial group theory, ...). With the desired results established in the group case, I will then consider the generalisations to the cases of partial actions of monoids and of so-called 'weakly left E-ample semigroups'. These latter semigroups arise naturally as subsemigroups of partial transformation monoids which are closed under a certain unary operation. We will see that the notion of 'globalisation' must be modified if we are to obtain such results for the partial actions of weakly left E-ample semigroups.

A Very Brief Introduction to Semigroup Theory

Graduate seminar, Department of Mathematics, University of York, 25th October 2007

Semigroup theory is a thriving field in modern abstract algebra, though perhaps not a very well-known one. In this talk, I will give a brief introduction to the theory of algebraic semigroups and hopefully demonstrate that it has quite a different flavour to that of group theory. In the first part of the seminar, we will build up some basic semigroup theory (with the emphasis on examples), though always with the group analogy at the back of our minds. Then, once we have established enough theory, we will break free of this restriction and see some truly 'independent' semigroup theory. Towards the end of the seminar, we will consider the application of semigroups to the study of 'partial symmetries'.

Yet Another Seminar Involving The 'P' Word

Graduate seminar, Department of Mathematics, University of York, 24th May 2007

The notion of a 'partial action' first appeared in the mid-90s in the context of operator algebras, specifically, in the guise of partial actions of groups on C*-algebras. Underlying such a partial action is that of a group on a set, the concept of which has received extensive study in recent years and, in particular, has spawned a generalisation to the monoid case.

Given the partial action of a group on a C*-algebra, however, we can also generalise in a different way: by stripping the C*-algebra of its 'analytical' structure and considering instead the partial actions of groups on associative algebras. Such study has proved extremely fruitful and has seen connections forged with the theory of partial group representations.

One of the goals of the study of partial actions of groups on C*-algebras was the construction of new C*-algebras by means of so-called 'crossed products'. This construction carries over to the purely algebraic study of the partial actions of groups on associative algebras in the form of the 'skew group algebra'. Conditions (on the underlying partial action) have been determined for when the skew group algebra is associative.

Given the partial action of a group on an associative algebra, we can also pose the perennial question: given such a partial action, when/how can we construct an action? One such method for constructing an appropriate action (termed the 'enveloping action' of the original partial action) has appeared in the literature, together with conditions for when such a construction is possible.

In this seminar, I will survey one of the major papers in this area: that of Dokuchaev and Exel (2004). I will discuss the relevant definitions and consider the associativity of the skew group algebra, before moving on the notion of enveloping actions. Towards the end of the talk, I will briefly indicate the connections between these concepts and that of a partial group representation. By way of conclusion, I will suggest possible generalisations of these ideas to the monoid case.

Partial Group Actions and Their Generalisations

Pure mathematics seminar, Department of Mathematics, University of York, 17th May 2007

The notion of the partial action of a group on a set first arose in the context of operator algebras and has since found a use in a wide range of other areas: model theory and tilings, for example. One question which has received particular attention is the following: given a partial group action, (when) can one construct an action? In this seminar, I will begin by discussing the definition of a partial group action (with examples), before describing one such method for constructing an action from a partial action: the process of 'globalisation'. This is a construction which appears in a number of other contexts (topology, combinatorial group theory, ...). With the desired results established in the group case, I will then consider certain natural generalisations.

The Prefix Expansion of a Monoid

Seminário de álgebra, Centro de Álgebra da Universidade de Lisboa, 17th November 2006

I will begin by defining the concept of an expansion, in the sense of Birget & Rhodes (1984), before moving swiftly to a detailed discussion of a specific expansion: the prefix expansion. In the study of non-regular semigroups, the prefix expansion has been largely over-shadowed by the related Szendrei expansion (Szendrei 1989). I will remedy this situation by demonstrating necessary and sufficient conditions for the prefix expansion of a monoid to be weakly left ample. (These conditions are analogous to those already obtained for the Szendrei expansion by Fountain, Gomes & Gould (1999).) I will next present a number of (surprisingly natural) examples of monoids satisfying these conditions. Finally, I will obtain, as a corollary to the 'weakly left ample' results, conditions for the prefix expansion to be inverse.

Partial Actions of Monoids, or How many times can the word 'partial' be used in the space of an hour?

Graduate seminar, Department of Mathematics, University of York, 19th October 2006

I will begin by discussing the concept of a partial group action and then, once the audience has been lulled into a false sense of security, it will be time to roll out the monoids. I will then build up the definition of a partial monoid action and give some justification as to why the definition we arrive at is the correct one. With the correct definition established, I will next introduce a little algebraic machinery which will enable me to present two distinct methods for constructing a (global) monoid action from a partial action. These methods generalise results from the group case. Unusually for me, there might even be some examples and applications.

Automata for Beginners

Graduate seminar, Department of Mathematics, University of York, 13th July 2006

The theory of automata first arose in the early 1950s, motivated in part by a desire to model mathematically the functioning of nerve cells. In the 50-ish years since then, automata have found applications in a wide variety of areas, for example: computer science, linguistics, combinatorial group theory... In this seminar, I will explore some of the basic ideas of both the theory of automata and of the theory of formal languages, to which automata theory is inextricably linked. I will demonstrate the fundamental connection between an automaton and a language, and also introduce some of the attendant finite semigroup theory.

A Discontinuous History of Mathematics

Graduate seminar, Department of Mathematics, University of York, 27th April 2006

The history of mathematics is a very long story, stretching back to the beginnings of civilisation. The mathematics of the truly ancient civilisations, such as the Sumerians, the Babylonians and the Egyptians, to name but a few, tended to be rather practical in nature. I will therefore start the story with the birth of the first pure mathematics in ancient Greece. From there, my (rather ambitious) intention is to cover the entire history of mathematics in an hour. Of course, this seminar cannot be comprehensive, so I hope to tell the story of mathematics via a sequence of snap-shots down the centuries, before bringing us finally to the present day.

Categorically a Seminar on Semigroups

Graduate seminar, Department of Mathematics, University of York, 9th March 2006

I will begin with a brief introduction to categories, giving an overview of the modern notion of a category as a generalised monoid. Then, with a little category theory established, I will introduce a particular type of category, called an inductive groupoid, and demonstrate its underlying connection with inverse semigroups, as described in the Ehresmann-Schein-Nambooripad Theorem. I will also explain just why it is that two such different-looking objects should share this fundamental connection. If time permits, I will explain, with the aid of a commutative diagram of epic proportions, how these results relate the study of actions and partial actions of both inverse semigroups and inductive groupoids.

Partial Actions of Right Cancellative Monoids

Graduate seminar, Department of Mathematics, University of York, 3rd November 2005

I will begin by giving a brief survey of existing results on the partial actions of groups on sets, before discussing possible definitions for the partial action of a monoid on a set, particularly in the very 'group-like' case of a right cancellative monoid. I will then introduce the so-called Szendrei expansion of a monoid and show that the Szendrei expansion of a right cancellative monoid is a left ample monoid. Using this, I will present a theorem which allows us to construct a full action from a given partial action of a right cancellative monoid.

Sums of Squares

Graduate seminar, Department of Mathematics, University of York, 27th January 2005

I will discuss the identity (a2 + b2)(c2 + d2) = (ac + bd)2 + (ad - bc)2 which expresses the product of two sums of two squares as the sum of two squares, and then describe the attempts that were made to find similar identities for sums of other numbers of squares, notably those for four and eight squares. I will go on to present a proof which tells us precisely when such an identity exists.