Taking the scenic route through Mathematics: memories of Ronnie Brown

Professor Tim Porter FLSW

(The following is adapted from an obituary to be published in the journal Cahiers de Topologie et Géométrie Différentielle Catégoriques, which will include a full list of Ronnie’s publications, together with more detailed discussion of his mathematical research and other activities.)

Ronnie Brown was born on 4 January 1935 to parents who were first generation emigrés to the U. K. from Romania. Ronnie went to Alleyn’s School, Dulwich, London, and then gained a place in New College, Oxford, in 1953. In Oxford, he met Margaret, also a student of mathematics. They married in 1958, and later had eight children.

After completing his undergraduate degree at Oxford, Ronnie went on to obtain his D.Phil., in 1961, again at Oxford in the then emergent area of Algebraic Topology. The subject of his thesis, on spaces of functions between topological spaces, showed several aspects of Ronnie’s future research interests, not only in algebraic topology, but also in what is now seen as part of the categorical side of topology. Many years later, when Ronnie was attending an international category theory meeting, he was surprised to discover that these, his first two papers, were considered to be of foundational importance for a large area of categorical topology.

During the period 1959 – 1964, Ronnie was first an Assistant Lecturer, and then Lecturer, at Liverpool University. He then became, from 1964 to 1970, a Senior Lecturer, then Reader, at the University of Hull. In 1968, he published a text book, ‘Elements of Modern Topology’, which was to become very influential, both for the teaching of the subject, and on the direction that Ronnie’s own research took subsequently. That book was revised several times, gaining additional sections as the subject matter evolved further.

It was while writing that book, and, as he said later, ‘to clarify certain points relating to the calculation of the fundamental group of a circle’, that he started taking an interest in the objects called groupoids and in particular in a groupoid version of a famous theorem of van Kampen, which was crucial for calculating various invariants of spaces as well as having important implications in some areas of algebra. The proofs of his new extended version suggested to him that there might be some interesting further higher dimensional extensions of that theorem and the search for these was one of the central themes of Ronnie’s research for much of his life.

In 1970, Ronnie moved to North Wales to take up the post of Professor of Pure Mathematics at the then University College of North Wales, part of the University of Wales. The family moved to Anglesey, to Benllech, and to a house just a short distance from the beach.

From 1974 onwards, Ronnie started exploring the topic of higher dimensional analogues of the groupoid van Kampen theorem, working with Phil Higgins, initially from King’s College London, later at Durham University. The main part of this research, lasting over 20 years, culminated in a large book, Non-abelian Homotopy Theory with Phil Higgins and Rafael Sivera. This explored the interactions of that new theory with other more classical themes. It also involved some ideas, especially those around structures known as crossed modules and what became known as crossed complexes, which had been developed initially by Henry Whitehead in the 1940s and 50s from ideas of both Whitehead and the German topologist, Reidemeister, in the 1930s, but whose study and use had lain fallow in the intervening period.

For Ronnie, there was a very interesting spin off from the study of crossed modules and crossed complexes. With collaborators, he examined pre-war work by Reidemeister on the area known as combinatorial group theory, which deals with ways of presenting the algebraic objects, known as groups, using combinatorial data. That work has had a profound impact on that subject area.

Further collaboration, (1983-87), again within this general area of ‘higher dimensional group theory’, occurred in collaboration with Jean-Louis Loday from Strasbourg, and centred on his models for what are called homotopy n-types, which correspond to truncating certain invariants of spaces above some (possibly high) dimension. Via a van Kampen style theorem for these models, the collaboration revealed a new type of product-like construction available when two (possibly non-commutative) groups act on each other in a compatible way. This lead to further collaborative work on more purely group theoretic aspects of this theory.

Although that area of ‘higher dimensional algebra’ was an important theme of Ronnie’s research, from about 1975, he also continued to work on more purely topological and geometric themes, extending ideas of two French mathematicians, Ehresmann and Pradines, to results with a more analytical geometric aspect.

Then in 1982, Ronnie wrote a letter to the famous French geometer, Alexander Grothendieck, and this started a very fruitful and amicable exchange of letters that lasted until 1991. His motivation had been to ask Grothendieck about his interest in notions of ∞-groupoids and ∞-categories, which were higher dimensional versions of the groupoids and categories that he had been using for many years. Certain types of these ∞-groupoids corresponded to the crossed complexes that Ronnie had been studying in detail for some time. This correspondence motivated Grothendieck to follow up on some ideas he had had some years earlier and to start on the famous long, but very influential, set of typed notes known as Pursuing Stacks, published in book form only in 2022, but initially distributed from Bangor with Grothendieck’s permission, as a photocopy of the original typescript. The models for (some) spaces that Ronnie was putting forward were strict ∞- groupoids and categories, corresponding to crossed complexes in one of their manifestations. They thus did not fully answer Grothendieck’s wider conjectural programme, which needed some weaker form of ∞-groupoids to handle all homotopy types. They were however seen to represent an important intermediate stage between the classical algebraic topological methodology and the vision of the theory proposed by Grothendieck that aimed at modelling all spaces by means of some form of weak ∞-groupoids.

We will return to Ronnie’s research interests later on, but now must consider his impact on other aspects of the subject area, and here it would not be correct to omit either his work in popularisation or on teaching innovation.

Ronnie’s involvement in the popularisation of mathematics is well known. In about 1985, with a group of colleagues and with the assistance of local schools in North West Wales, he initiated a series of Mathematical Masterclasses for Young People in the area, under the aegis of the Royal Institution of Great Britain. This led to the development in Bangor of material that was designed to help in some of those Masterclass sessions, and also to the preparation of a set of exhibition boards on the theme of ‘Maths and Knots’ for use when giving talks in the masterclasses, and, increasingly, elsewhere.

Very quickly, this project was to acquire another component. As Ronnie put it: ‘One day in May 1985, I was walking down Albermarle Street from a meeting on masterclasses at the Royal Institution. As I passed the Freeland Gallery, … and with some time to spare, I decided to wander inside, enticed by the sculptures of children and animals shown in the window. To my amazement I found also some strong and beautifully crafted knot sculptures’. Thus started a collaboration and friendship between Ronnie and the sculptor, John Robinson, and a widening out of the scope of Ronnie’s popularising work on the theme of ‘How Mathematics gets into Knots’. This led to Bangor’s involvement in the Pop Maths Roadshow, and in a pan-European project on Raising Public Awareness of Mathematics for European National Science Week, 2000. This involved lots of fun interaction with lots of interesting people … and a lot of thought and hard work by the team in Bangor!

Partly as a result of working with the younger students in the masterclasses, and with the general public through popular lectures and the exhibition, Ronnie, with myself and others, started thinking about the context of the mathematics that was being taught to our own students. We realised, for instance, that even our own students were not really aware that new mathematics was being discovered / created all the time, nor how that was done. To counter this, we developed what was a new type of course to form part of the mathematics degree. This was called ‘Mathematics in Context’. The style was very informal. We discussed historical, cultural and scientific issues, and got very good and interesting input and feedback from the students. The sessions were very enjoyable and were, in general, very well received by both the students and, when the assessment was completed, by the external examiners for the degree.

The idea of adding ‘context’ explicitly into courses interacted with discussions on the methodology of mathematics within the sessions. Printed versions of some of those discussions were published in journals in various countries and languages as they seemed to strike a useful note with many people.

This emphasis on methodology, and on making research more approachable, started to feedback into Ronnie’s own research. It meant that he had a very clear idea on how to explain methodological ideas to non-mathematicians, partially bridging the well known gap between the mathematical view and the viewpoints of other scientists.

Running through Ronnie’s research, there are some themes that recur, and these also pervade other branches of science. One such is the idea of ‘local to global’, so how ‘local’ information about an object coalesces to give global information. Another, which is almost the converse of that, was the idea that ‘subdivision is an inverse to composition’. For Ronnie, the local to global paradigm was exemplified by the various higher algebraic structures that grew out of the quest for generalisations of van Kampen’s theorem. That local to global aspect, however, also provided insights into various philosophical and scientific questions, which were illustrative of the methodology of mathematics, for example questions of abstraction, modelling geometric situations via algebra, and the development of new concepts.

In discussions, he realised how some of these paradigmatic problems were important in some other very interesting areas. In computer science, for instance, the local to global problem interacted with ideas on concurrent computing, whilst ideas on ‘composition versus subdivision’ led to discus- sions on modelling neurosystems, and more general hierarchical systems in biology and biocomputing. Philosophical / psychological contexts in which the problems of abstraction and the refinements of concepts occurred also mirrored quite closely those encountered in mathematics. These discussions led to collaboration with researchers on biocomputing and computational biology.

Throughout his life as a research mathematician, Ronnie served in editorial roles for international journals. From 1975 to 1994, he was on the editorial advisory board for the London Mathematical Society. He was a founding member of the editorial board of Theory and Applications of Categories, then on the editorial board of Applied Categorical Structures. In 1999, he helped found the electronic journal, Homology, Homotopy and Applications and then, from 2006, was an editor for the Journal of Homotopy and Related Structures. In 2016, a special volume of that journal was dedicated to him and his work on the occasion of his 80th birthday.

Ronnie retired from full time teaching in 1999, although he continued as a half-time research professor until 2001. In 2016, Ronnie was elected to Fellowship of the Learned Society of Wales and he was a lifelong member of the London Mathematical Society.

By about 1998, Ronnie had become hard of hearing on one side, and during a visit to Bielefeld for discussions with Tony Bak, he suffered from a severe loss of balance. On return to North Wales, he was diagnosed as having an acoustic neuroma, which is a non-cancerous growth covering the acoustic nerve. He was treated for this in February 2000, and for a time was feeling a bit better. He then began to suffer from side effects of the treatment of the neuroma. These were successfully handled in 2001, and he gradually recovered and continued to work on his research projects.

Although he and Margaret had eight children, throughout his life he had found time to play firstly table tennis, then later on squash and to go swim- ming, both in the sea and in the lakes of North Wales. He also spent time gardening, making home made beer, searching for mushrooms in season, and exploring the region with the children when they were young.

On retirement, he and Margaret had moved from Anglesey to Deganwy, still in North Wales, and to a house with a beautiful view westward along the coast. They also had a small cottage for family reunions, and went on several Mediterranean cruises, including one during which he was roped in to explain some aspects of Greek mathematics! They both participated in lots of the cultural life of the area.

Ronnie’s wife, Margaret, died in 2020, and Ronnie’s mobility had become reduced after a stroke. He was cared for by their eldest son, and enjoyed excursions using a mobility scooter, including to see seals in a cove a short drive away from their home, to visit a nature reserve on the Conwy estuary or the gardens at Bodnant, a short distance south along the Conwy valley. It was after one of the visits to see the seals that he passed away peacefully, but suddenly, at home in Deganwy, aged 89 years. He will be missed by his seven surviving children, and his grandchildren.

Perhaps the last word should be left to his children who said that on journeys, their father would deviate from the main roads to “take the scenic route” – sometimes getting lost in the process. But his attitude to life was always “take the scenic route”. That love of exploring ideas or places for their own sake permeated his life both in his research career and in the teaching that he loved.

Tim Porter, FLSW, May 2025