# Books

*Beyond the Academy: The Practice of Mathematics 1600–1850* (co-edited with Philip Beeley), to be published by Oxford University Press.

The tremendous growth of the mathematical sciences in early modern Europe was reflected contemporaneously in an increasingly sophisticated level of practical mathematics in fields such as merchants’ accounts, instrument making, teaching, navigation, and gauging. Mathematics in many ways shaped the knowledge culture of the age, extending through the Industrial Revolution to the nineteenth century. While theoretical developments in the history of mathematics have been made the topic of numerous scholarly investigations, in many cases based around the work of key figures such as Huygens, Leibniz or Newton, practical mathematics, especially from the seventeenth century onwards has been largely neglected. This volume aims to cover the full range of mathematical practice from the seventeenth century through to the middle of the nineteenth century. It coincides with increased historical interest in the social and cultural milieus in which pre-modern and modern science were carried out, and facilitates and promotes further investigations on the practice of mathematics itself.

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 book focuses upon and stresses the historically unique character of the Oslo congress 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. Relying heavily on unpublished archival sources, we consider the differences between the goals of the various participants in the congress, most particularly the Norwegian organisers, and the Nazi-led German delegation. We consider also the background to the absence of the proposed Soviet and Italian delegations. In addition, we go into the mathematical dimension of the Oslo congress. This was the conference at which the Fields Medals were awarded for the first time, and we put the laudatory addresses given on this occasion into perspective. We give overviews of the 19 plenary presentations and their planning and development, and add biographical information about each of the plenary speakers. We also put the state of international mathematical communication beyond narrow political conditions into perspective; the Oslo congress is used as a lens through which to view the state of the art of mathematics in the mid-1930s.

*Ada Lovelace: the making of a computer scientist* (with Ursula Martin and Adrian Rice), Bodleian Library Publishing, 2018; distributed in North America by The University of Chicago Press; Spanish translation by Tayra Lanuza: *Ada Lovelace: La formación de una científica informática*, Publicacions de la Universitat de València, 2019

Ada Lovelace (1815–1852) is hailed as the world's first computer programmer and has become an icon for women in science. In this book, we tell the story of her mathematical education, leading up to her famous collaboration with the computer pioneer Charles Babbage: for Lovelace and Babbage computing was a branch of mathematics. The story is told through the reproduction (with brief commentaries) of selections from her mathematical papers and correspondence, held in the Bodleian Library.

This short book features a translation into English of the 1953 paper 'Theory of generalised heaps and generalised groups' by the Russian mathematician V. V. Wagner (1908–1981), with the goal of bringing his elegant theories concerning abstractions of systems of binary relations to a wider readership. We provide historical and mathematical context for these ideas, and point to their legacy in present-day mathematics, in the hope of sparking new lines of enquiry.

Drawing evidence from a range of disciplines, I study the extent to which scientists were able to communicate with their counterparts on the opposite side of what became the Iron Curtain. I consider the scope that existed for personal communication between scientists, as well as for the attendance of foreign conferences, and describe how these changed over the decades. Access to publications is also dealt with: I address separately the questions of physical access, and of linguistic access. In particular, I argue that physical accessibility was generally good in both directions, but that Western scientists were afflicted by greater linguistic difficulties than their Soviet counterparts, whose major problems in accessing Western research were bureaucratic in nature.

A semigroup is a set which is closed under an associative binary operation. It therefore represents an abstraction of the system of all self-mappings of a set, in much the same way that a group provides us with an abstract version of a system of permutations. The theory of semigroups is a relatively young branch of mathematics, with most of the major results having appeared after the Second World War. In this book, I describe the evolution of semigroup theory from its earliest origins to the establishment of a fully-fledged theory. The study of semigroups encompasses a wide range of topics, and much of it may be organised under the headings of 'algebraic semigroup theory' and 'topological semigroup theory'. Although these two are not mutually exclusive, the focus here is upon the development of the algebraic side of the theory.

Because of the time during which it developed, semigroup theory might be termed 'Cold War mathematics': there were thriving schools on both sides of the Iron Curtain, although the two sides were not always able to communicate with each other, or even gain access to the other's publications. One of the major themes of this book is the comparison of the approaches to the subject of mathematicians in East and West, and the study of the extent to which contact between the two sides was possible. I consider the way in which parallel developments in East and West shaped the subsequent theory of semigroups.

At the start of the book, some background is given on the abstract algebra of the early twentieth century, and on communications difficulties across the Iron Curtain. I then begin the discussion of the development of semigroup theory by considering the work of a Russian pioneer, A. K. Sushkevich, before moving on to look at certain semigroup-theoretic problems that emerged in the 1930s by analogy with similar problems for rings. Around the mid-point of the book, I describe the derivation of semigroup theory's first major structure theorems (around 1940), which might be taken as marking the beginning of a truly independent theory of semigroups. Thereafter, I indicate the ways in which the theory expanded in the following decades, both in terms of the topics considered and also through its internationalisation. The major themes of the later theory are outlined. The book concludes with an investigation of the early books, seminars and conferences on semigroups. In particular, I discuss the first international conference (Czechoslovakia, 1968), which led to the foundation of a journal devoted exclusively to semigroup research.