 "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. x 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. x  Language use in Soviet mathematics journals
History of mathematics minisymposium, 4th Joint BMC/BAMC, University of Cambridge, 31st March 2015. x 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. x  Soviet participation in the ICMs: the early stages of a study
Mathematics: place, production and publication, 17301940, Institut MittagLeffler, Stockholm, 26th January 2015. x A short account of work in progress. x  Russian views of English algebra: a preliminary study Slides
Short talk to Oxford History of Mathematics Group, The Queen's College, Oxford, 26th May 2014. x 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. x  Mathematics across the Iron Curtain Slides
BSHM Christmas Meeting, The Birmingham and Midland Institute, Birmingham, 7th December 2013. x
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'. x  Mathematics across the Iron Curtain Slides
Talk to Oxford History of Mathematics Group, The Queen's College, Oxford, 14th October 2013. x
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'. x  An Obsession for Documentation: Surveys of Mathematical Progress in the USSR Slides
History of mathematics workshop, 65th British Mathematical Colloquium, University of Sheffield, 27th March 2013. x 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. x  Densely Embedded Ideals: A Handy Tool from the Pages of Soviet Semigroup Theory Slides
York Semigroup, Department of Mathematics, University of York, 10th December 2012.
x 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. x  Unique Factorisation from Rings to Semigroups Slides
Algebra and geometry seminar, School of Mathematics, University of Bristol, 5th December 2012. x 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. x  Adventures in Kharkov Slides
Talk to Oxford History of Mathematics Group, The Queen's College, Oxford, 28th May 2012. x 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. x  Левые рестриктивные полугруппы (Left restriction semigroups)
66th Herzen seminar (algebra section), Herzen Russian State Pedagogical University, St. Petersburg, 20th April 2012. x Я буду говорить о левых рестриктивных полугрупп и об их связи с левыми индуктивными созвездиями. x I will talk about left restriction semigroups and their connection with left inductive constellations. x  The Perils of Taking Shortcuts: Embedding Semigroups in the 1930s Slides
NBSAN, School of Mathematics, University of Manchester, 30th August 2011. x
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. x  A. H. Clifford and Unique Factorisation: Some Forgotten Early Steps into Semigroup Theory Slides
Groups and Semigroups: Interactions and Computations, Faculdade das Ciências, Universidade de Lisboa, 26th July 2011. x 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. x  A Short History of Inverse Semigroups Slides
York Semigroup, Department of Mathematics, University of York, 16th February 2011. x 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. x  Beyond the EhresmannScheinNambooripad Theorem Slides
York Semigroup, Department of Mathematics, University of York, 25th May 2010.
x 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. x
 The Early Days of Semigroup Theory, or Learning to be an Historian Slides
Talk to Oxford History of Mathematics Group, The Queen's College, Oxford, 26th April 2010.
x
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.
x
 Enveloping Actions for Partial Actions on Rings
Seminário de álgebra, Centro de Álgebra da Universidade de Lisboa, 23rd October 2009.
x
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'.
x
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.
x
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.
x
 Sistemas de algarismos nãoposicionais Slides
Talk to 'Cinfinito': Clube de Matemática da FCUL, 1st October 2009.
x
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.
x
 An ESN Theorem for ∧premorphisms Slides
Workshop on Groups, Semigroups and Applications, Centro de Álgebra da Universidade de Lisboa, 24th April 2009.
x
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.
x
 Partial Actions of Inverse Monoids on KRings
Miniworkshop on algebra, Centro de Álgebra da Universidade de Lisboa, 18th April 2008.
x
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.
x
 Partial Actions of Inverse Monoids on KRings
Semigroup splinter group, 60th British Mathematical Colloquium, University of York, 26th March 2008.
x
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.
x
 Partial Group Actions and Their Generalisations
Seminário de álgebra, Centro de Álgebra da Universidade de Lisboa, 30th November 2007.
x
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.
x
 A Very Brief Introduction to Semigroup Theory
Graduate seminar, Department of Mathematics, University of York, 25th October 2007.
x
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'.
x
 Constructing Incomplete Actions Slides
Conference on Semigroup, Acts and Categories, with Applications to Graphs, Tartu Ülikool, Estonia, 29th June 2007.
x
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.
x
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.
x
 Yet Another Seminar Involving The 'P' Word
Graduate seminar, Department of Mathematics, University of York, 24th May 2007.
x
 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.
x
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.
x
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.
x
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.
x
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.
x
 Partial Group Actions and Their Generalisations Slides
Pure mathematics seminar, Department of Mathematics, University of York, 17th May 2007.
x
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.
x
 The Prefix Expansion of a Monoid Slides
Seminário de álgebra, Centro de Álgebra da Universidade de Lisboa, 17th November 2006.
x
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.
x
 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.
x
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.
x
 Partial Actions of Monoids Slides
Fountainfest, University of York, 14th October 2006.
x
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.
x
 Automata for Beginners
Graduate seminar, Department of Mathematics, University of York, 13th July 2006.
x
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.
x
 A Discontinuous History of Mathematics
Graduate seminar, Department of Mathematics, University of York, 27th April 2006.
x
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.
x
 Categorically a Seminar on Semigroups
Graduate seminar, Department of Mathematics, University of York, 9th March 2006.
x
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.
x
 Partial Actions of Right Cancellative Monoids
Graduate seminar, Department of Mathematics, University of York, 3rd November 2005.
x
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.
x
 Sums of Squares
Graduate seminar, Department of Mathematics, University of York, 27th January 2005.
x
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. 
