**Otakar Boruvka**entered the Gymnasium in Uherské Hradiste in 1910 when he was eleven years old. He continued to study there until 1916 but, of course, by this time Europe was in the midst of World War I. The war was unpopular with the Czech people, and the press was censored and public meetings were forbidden. With the influence of the military increasing, Boruvka went to the military school in Hranice in northern Moravia. He then moved to another military school in Mödling in the picturesque Brühl Valley, just southwest of Vienna.

After the end of World War I, Boruvka returned to the Gymnasium in Uherské Hradiste to take his final examinations. He then entered the Czech Technical University in Brno where he studied from 1918 to 1922. Here he came under the influence of Mathias Lerch who, after eight years as professor at the University of Fribourg in Switzerland, had returned to the Czech Republic in 1906 when he was appointed professor of mathematics at the Czech Technical University in Brno. In 1920 Boruvka became an assistant at the Institute of Physics and, in the same year, began taking courses also at the Masarak University in Brno. This was a new university, opened in 1920, named the Masaryk University after the first president of Czechoslovakia, Tomás Masaryk. Lerch became the first professor of mathematics at this new university and he selected Boruvka to become his research assistant in 1921. Sadly the collaboration did not last long since Lerch died in August 1922. Eduard Čech was appointed extraordinary professor at Masaryk University in 1923 to fill the vacancy created by Lerch's death. Boruvka then became Čech's research assistant and Čech interested Boruvka in differential geometry. After being awarded his doctorate in 1923, Boruvka continued to undertake research at Masaryk University.

To many people Boruvka is best known for his solution of the Minimal Spanning Tree problem which he published in 1926 in two papers *On a certain minimal problem* (Czech) and *Contribution to the solution of a problem of economical construction of electrical networks* (Czech). Let us quote the problem as it appears in the second of these 1926 papers:-

In modern graph theoretical terms this can be stated as: Given an undirected graph with weights assigned to its edges, find a spanning tree of minimal weight.There are n points in the plane whose mutual distances are different. The problem is to join them with a net in such a way that:1

. any two points are joined to each other either directly or by means of some other points;2

. the total length of the net will be minimal.

In fact the problem had been suggested to Boruvka before he became a university student. He had a friend, Jindrich Saxel, who worked for the firm West-Moravian Powerplants and he suggested the problem which he stated in terms of cities and the distances between them. At the time that Saxel suggested the problem to Boruvka, World War I was still happening and Czech universities were closed. Boruvka was offered a job with West-Moravian Powerplants at this time but declined. The authors of [8] write:-

Čech suggested to Boruvka that he should go abroad and broaden his mathematical horizons. An obvious choice, suggested Čech, was Paris where Boruvka could work with Élie Cartan. He spent 1926-27 in Paris, where he lectured on his solution to the Minimal Spanning Tree problem, then returned to Masaryk University in Brno where he habilitated and was made a dozent in 1928. He spent further years abroad, going back to Paris in 1929-30 where his visit was supported by the Rockefeller foundation. He then went to Hamburg where, still supported by the Rockefeller foundation, he spent the year 1930-31.The Minimal Spanning Tree problem is a cornerstone of Combinatorial Optimisation and in a sense its cradle. The problem is important both in its practical and theoretical applications. Moreover, recent development places Boruvka's pioneering work in a new and very contemporary context. One can even say that out of many available Minimal Spanning Tree algorithms, Boruvka's algorithm is presently the basis of the fastest known algorithms.

The graph theory work undertaken by Boruvka early in his career did not lead to further work in this area. Under the influence of Eduard Čech and Élie Cartan he worked on differential geometry, then he became interested in algebra, and undertook research on groups and groupoids (algebraic systems in which the associative law does not hold). For example in *Über Ketten von Faktoroiden* Ⓣ (1941) he considered partitions of groupoids with the property that the product of two sets is contained entirely within some other set of the partition. In 1944 he published a little book of 80 pages *Introduction to the Theory of Groups* (Czech). The book consists of three chapters: the first gives the fundamental concepts of set theory; the second contains the theory of groupoids; and the third contains the theory of groups. The book went through a number of editions, each adding considerably more material. The second Czech edition in 1952 contains about 150 pages, while when the book was published in German in 1960 it was about 200 pages long. Its aim in presenting material in both groups and groupoids was reflected in the title *Grundlagen der Gruppoid- und Gruppentheorie* Ⓣ. Essentially the material of this German edition appeared in Czech in 1962 and in English in 1976 with the title *Foundations of the theory of groupoids and groups*.

As World war II was drawing to a close, Boruvka began to think about revitalising the mathematical research work of Masaryk University. He discussed these matters with Frantisek Vycichlo, a Prague mathematician, and their feeling was that differential equations would be a good direction to take his research team. Boruvka had already written a paper on differential equations in 1934, but now he began to direct the research of Masaryk University towards that topic. His own recollections about this period are given in [22]:-

In 1946 Boruvka became an ordinary professor at Masaryk University and in the following year he set up a Differential Equations Seminar. The main aim of the seminar was to study global properties of linear differential equations of theAlready in1944, when it was clear that the war would be over soon and that the victory of the Allies was certain, it was necessary for me to think about my future activity realistically, i.e. on the one hand my pedagogical activity, but also my scientific one as well. As for the pedagogical activity, it was necessary to see to it that those students who started studying before the war might accomplish their studies and the newcomers might begin studying ... How to organize lectures to satisfy all those students? In this respect I did not worry ... I was rather thinking of what trend to start, as far as scientific work was concerned. At that time scientific work was not controlled in any way and the professors bore the responsibility personally and individually and I had not a good overview of what it looked like in this country on the whole and in what direction scientific work in mathematics should continue. I went to Prague, it was at the end of1944, to consult the matter with my colleagues. I spoke above all to Frantisek Vycichlo whom I held in high esteem. We discussed the matter thoroughly and arrived at the conclusion that it was essential to start pursuing the theory of differential equations which is immensely important as far as applications are concerned and which was much neglected before the war and in essence it was not at all developed. And since we did not see anyone that would take up that task, I declared that I would take the matter myself, although it was not an easy decision, because it meant I had to change the field of my scientific work.

*n*th order. He explains [22] how he went about this work:-

Boruvka's publications on this topic includeVery soon I recognised that it was an immensely difficult problem of long lasting which I could not master with my own forces in the near future. The main problem and difficulty was the fact that here there occurred absolutely new questions which had no models, no basic concepts were known, not to mention any methods that would permit some systematic study, etc. And that is why I came with the idea that the solution of that problem was possible only in the following way that in the first period one would acquire some experience in the simplest cases and only in the second period, on the basis of the concepts introduced and experience obtained, one would go to the extension of those results to the most general case. And I did it in that way.

*Sur les intégrales oscilatoires des equations différentielles linéaires du second ordre*Ⓣ (1953),

*Remark on the use of Weyr's theory of matrices for the integration of systems of linear differential equations with constant coefficients*(Czech) (1954),

*Über eine Verallgemeinerung der Eindeutigkeitssätze für Integrale der Differentialgleichung*

*y*' =

*f*(

*x*,

*y*) Ⓣ (1956), and

*Sur la transformation des intégrales des équations différentielles linéaires ordinaires du second ordre*Ⓣ (1956). Much of his work, and that of others participating in his seminar, is contained in Boruvka's book

*Lineare differential- transformationen 2. Ordnung*Ⓣ (1967) which was translated into English as

*Linear differential transformations of the second order*(1971).

Among the many honours which Boruvka receiced were election to the Czech Academy of Sciences (corresponding member 1953, ordinary member 1965), and honorary doctorates from Bratislava (1969) and Brno (1994).

Let us end this biography with Boruvka's own words about his approach to mathematics [22]:-

I would like to remember facts in my life that were essential not only for me personally, but chiefly for mathematics and for the future mathematical generation: Before every serious task I try to find carefully and dutifully how to fulfil it in the best way, and when I find a solution, I carry it out as best as I can according to my best sense and conscience and with all my might. I consider success a natural consequence of my activity and I do not ascribe it a particular importance. I consider failures to be signs of the complexity of life and I draw information from them. But I am never sorry for my decisions, because at every moment I acted as best as I could. ... And maybe just because none of us knows which day will be his or her last, I tried knowingly, and according to my powers, to live in each of them fully and to work. In the same way as my teachers lived - Matyas Lerch, Ladislav Seifert and Eduard Čech - they gave me a lot - I also feel the duty to pass most of it to the young talented generation. They always sided with the talented and diligent ones, that was their, and in the end also my, creed: I will set you on the horse, but you must ride the horse yourselves.

**Article by:** *J J O'Connor* and *E F Robertson*

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