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.
Mathematics at the Philosophical and Literary Societies
The founders of the Philosophical and Literary Societies that emerged across Britain in the lateeighteenth and earlynineteenth 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
The process of generalisation in mathematics sometimes gets a bad press owing to the presence in the literature of many an illmotivated `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 wellmotivated generalisation. I will give a survey of a selection of these, and point out some of their interesting interconnections.
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.
Skype lecture for Ada Lovelace Day, University of Edinburgh, 11th October 2016.
A brief summary of the education and scientific work of Ada Lovelace, with a particular focus on the figures who influenced her either as tutors or mentors.
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 EastWest 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 Englishspeaking Westerners and assess the effectiveness of scientific foreignlanguage teaching and the impact of the systematic translation of Soviet resources.
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 postSoviet 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
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 nativeEnglishspeaking mathematicians in the use of other languages. In the final part of the talk, I focus specifically on Englishspeaking Westerners and assess the effectiveness of scientific foreignlanguage teaching and the impact of the systematic translation of Soviet resources.
Meeting under the integral sign? The 1936 Oslo International Congress of Mathematicians
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 whistlestop tour of the early ICMs, before discussing the circumstances of the Oslo meeting, with a particular focus on the activities of the Naziled German delegation.
A brief history of semigroup representations
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.
R. S. Ball's 'Mechanics' and the 'London Science ClassBooks', or Bringing Newton to the masses?
The 'London Science Classbooks' 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 thenRoyal Astronomer for Ireland, Robert Stawell Ball.
I give a brief overview of the ups and downs of EastWest scientific communication during the twentieth century, and during the Cold War in particular.
Russian participation in the early International Congresses of Mathematicians
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 participation in the early International Congresses of Mathematicians
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.
During the years 18401, 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 indepth 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.
Ada Lovelace: 19thcentury mathematician and computer scientist
A short account of Ada's life, and an indication of why she is famous.
Soviet views of early (English) algebraJoint 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 (nonRussian) 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. 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 indepth 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.
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 studyMathematics: place, production and publication, 17301940, Institut MittagLeffler, Stockholm, 26th January 2015. A short account of work in progress.
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.
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'.
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'.
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.
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.
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.
I will give a biographical sketch of the Russian algebraist A. K. Sushkevich (18891961). 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.
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.
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.
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.
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.
The celebrated EhresmannScheinNambooripad 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 EhresmannScheinNambooripad Theorem involving the category of inverse semigroups and ∧premorphisms.
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.
A partial action is, loosely speaking, an action which is not everywheredefined. 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 socalled '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.
Com muitos exemplos da história, falarei sobre os sistemas de algarismos em geral, e os sistemas nãoposicionais em particular. Examinarei o desenvolvimento destes sistemas e compararei sistemas diferentes.
With lots of examples from history, I will speak about systems of numerals in general, and nonpositional systems in particular. I will examine the development of these systems and compare different systems.
The EhresmannScheinNambooripad (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 ESNtype theorem for inverse semigroups and ordered ∧premorphisms.
The partial actions of groups on Krings (a.k.a. associative Kalgebras) 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 Krings and describe the construction of enveloping actions for such partial actions.
The partial actions of groups on Krings (a.k.a. associative Kalgebras) 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 Krings and describe the construction of enveloping actions for such partial actions.
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 socalled 'weakly left Eample 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 Eample semigroups.
Semigroup theory is a thriving field in modern abstract algebra, though perhaps not a very wellknown 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'.
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 Eample 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 Eample 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 Eample 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.
The notion of a 'partial action' first appeared in the mid90s 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 socalled '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.
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.
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 nonregular semigroups, the prefix expansion has been largely overshadowed 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.
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.
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.
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 50ish 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.
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 snapshots down the centuries, before bringing us finally to the present day.
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 EhresmannScheinNambooripad Theorem. I will also explain just why it is that two such differentlooking 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.
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 'grouplike' case of a right cancellative monoid. I will then introduce the socalled 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.
I will discuss the identity (a^{2} + b^{2})(c^{2} + d^{2}) = (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.

