**Olli Lehto** was educated in Helsinki and, as he grew up, he was pulled in different directions as to the subjects in which he should specialise. He was very fond of geography, and considered this might be the topic for him, but he also had a great passion for butterflies - this was his schoolboy hobby. However, there were other influences on the young boy [4]:-

I also considered going into the field of law. My uncle was a professor at Helsinki University of Technology and tried to lure me into the field of technology, but I knew all along that it wasn't for me. My father had a Master's degree in mathematics. I figured that the field would offer me an enormous range of possibilities after graduation.

Because World War II broke out, Lehto never took his matriculation examination and had to sit behind an artillery gun on Finland's eastern border. However, we need to look at some Finnish history in order to understand the events that took place during World War II. Finland had taken the opportunity of declaring its independence in December 1917 when Russia was occupied with its own internal affairs of the Russian Revolution. Nevertheless, the country continued to see its security threatened by the Soviet Union and, after a failed attempt to create an alliance with Estonia, Latvia, and Poland in the 1920s, Finland signed a non-aggression treaty with the Soviet Union in 1932. At the start of World War II in 1939, Russia and Germany had a pact, the so-called Ribbentrop-Molotov pact, to divide Poland between them. The two-pronged attack - the Germans from the west and the Russians from the east - quickly defeated the Polish army. The Soviet Union clearly did not trust their German allies, for they demanded territory from Finland to improve the security of Leningrad. When the Finns refused, the Soviet Union attacked Finland on 30 November 1939. The Finns put up a remarkably good defence against their massive neighbour but, after initial Finnish victories, they were forced to sign the Treaty of Moscow in March 1940 which leased large parts of south east Finland to the Soviet Union. Finland now saw Germany as a possible ally against the Soviet threat and, although no formal agreement was signed, German troops entered Finland. When Germany attacked their former ally, the Soviet Union, on 22 June 1941, Finland moved their forces to retake the land given up to the Soviet Union in the Treaty of Moscow.

Despite not having taken the matriculation examinations, Lehto was able to enter the University of Helsinki in 1943. He said [4]:-

1943

was the last year you could still 'sneak' into the university, without passing the matriculation examination.

However, he was better prepared than this quote might suggest, for he spent the summer before beginning his university studies reading on his own the materials for the first year courses.

German defeats, however, eventually led to the Soviet Union again taking land from Finland in June 1944 and in September of that year they agreed again to the terms of the Treaty of Moscow. This required German troops to leave Finland but they refused to do so. Finland now attacked the 200,000 German troops stationed in Lapland. This "War of Lapland" dragged on until 27 April 1945. Lehto was a soldier in the Finnish army training at the Riihimäki Artillery Training Academy in Riihimäki, about 70 km north of Helsinki. As well as fighting the Soviets on Finland's eastern border, he was also involved in the "War of Lapland" in the north where he fought against the Germans [5]:-

Privately, he likes to point out that in World War II he had the rare luxury to take a few shots at "both the Communists and at the Nazis".

It was only after World War II that Lehto became committed to mathematics. He writes in [1]:-

Mathematics hit me like a tsunami when I was almost exactly20years old, when I was able to get over the war years during which I had suppressed my desire to learn new things.

From that time on he was an outstanding student but his chance of an international education came through a piece of good fortune. During World War II, Rolf Nevanlinna had been rector of the University of Helsinki but, after the defeat of Germany in 1945, the authorities at Helsinki University felt that it would be inappropriate to have a rector who was known to have German sympathies so he was asked to resign from that role. He retained his chair at Helsinki, however, but in October 1946 he visited Zürich. This marks the beginning of a fifteen year period during which he was a guest lecturer in Zürich. Nevanlinna played an important role in Lehto's career beginning from the time that he was an examiner of Lehto's Master's thesis. Impressed with Lehto's work, Nevanlinna arranged for him to receive a scholarship to study with him at Zürich [3]:-

Without his help, I'd have had no possibility to go abroad. It was quite rare at that time.

His studies in Zürich proved particularly significant [4]:-

In Zürich, I was able to focus on my doctoral thesis and participate in the research seminar led by Nevanlinna. That was the moment when I noticed how inspirational teaching can be.

Lehto had been awarded his Master's degree in 1947, and he received his doctorate from the University of Helsinki two years later for his thesis *Anwendung orthogonaler Systeme auf gewisse funktionentheoretische Extremal- und Abbildungsprobleme* Ⓣ. A review of the 51-page thesis by Z Nehari indicates the areas that it covered:-

In Chapters1-3of this thesis, the author largely restates earlier results, due mainly to Bergman and Schiffer, concerning the application of complete sets of complex orthonormal functions to the theory of conformal mapping. Despite its great intrinsic elegance and its adaptability for numerical computations, the theory of complex orthonormal functions(centring about the concept of the Bergman kernel function)had the drawback of being a mere representation theory; the fundamental existence theorems had to be borrowed from other fields. In Chapter4the author fills this gap in one important instance by giving an existence proof for the parallel slit mappings(in the case of simply-connected domains this is identical with the Riemann mapping theorem)within the framework of the orthonormal function theory. Over and above its role in filling this gap, this proof is significant for being constructive and not(like the classical proofs)a mere "existence proof." The mapping function is first constructed in terms of a complete orthonormal set, and it is then shown that the obtained function has indeed the desired mapping properties.

Lehto published many works in the years following the award of his doctorate. For example, *On the existence of analytic functions with a finite Dirichlet integral* (1949), *On Hilbert spaces with a kernel function* (1950), *Some remarks on the kernel function in Hilbert function space* (1952), *On the distortion of conformal mappings with bounded boundary rotation* (1952), *A majorant principle in the theory of functions* (1953), *On an extension of the concept of deficiency in the theory of meromorphic functions* (1953), and *Value distribution and boundary behaviour of a function of bounded characteristic and the Riemann surface of its inverse function* (1954).

During the early part of his career, Lehto had two jobs, one as a mathematician at the cable company Suomen Kaapelitehdas Oy, a predecessor of Nokia, where he worked from 1947 to 1962. His other job was at Helsinki University where he was a docent from 1951 to 1956 and an associate professor from 1956 to 1961 when he was promoted to full professor. He held this position as professor until 1988. He served in a number of important roles during these years: he was Dean of Science from 1978 to 1983, Rector of the University from 1983 to 1988 and Chancellor from 1988 to 1993. In addition, he had various roles outside the University of Helsinki. For example he was Rolf Nevanlinna's assistant at the Finnish Academy of Science and Letters from 1953 to 1956. He was also a research professor at the Finnish Academy of Science from 1970 to 1975.

There are a number of different aspects of Lehto's career that we must emphasise. First there is his job with the cable company in the early part of his career. He described it as:-

A gainful job that I did with my left hand.

But, as is stated in [3]:-

... he ended up as the leading figure in the electronics department that was established by the company; that is, he contributed to building the foundation for Finnish computer science. Finland's first professors of the subject were trained at the cable company.

We have already indicated something of his contributions to function theory. A particular aspect that interested him was the theory of quasiconformal mappings. We illustrate this by quoting from a review by Lars Ahlfors of Lehto's text, written in collaboration with K I Virtanen, *Quasikonforme Abbildungen* Ⓣ (1965):-

From a seed planted by H Grötzsch in1928, the theory of quasiconformal mappings has developed a healthy life of its own that could not easily have been foreseen. Originally not much more than a curiosity, its notions began to pervade the theory of elliptic differential equations in two variables(Lavrent'ev, Morrey), and Teichmüller saw it not merely as an efficient tool in geometric function theory, but as a gateway to new problems of unmistakably classical flavour. It led in due time to a simple solution of the geometric problem of moduli, and there are encouraging signs of a fruitful theory in several dimensions. Quasiconformal mappings derive much of their fascination from the constant interplay between geometric and analytic methods. This is the leading point of view in this book, to the extent that the authors ignore all applications and even the higher-dimensional analogues. The book is thus intended for readers who are already convinced of the importance of quasiconformal mappings as an independent theory. It is written with a great deal of enthusiasm that will appeal to experts and non-initiates alike. The authors have a very fresh approach, and in almost all instances they improve on existing proofs. In this sense the book contains much original work.

Perhaps at this point we should also look at Lehto's monograph *Univalent functions and Teichmüller spaces* (1987). William Abikoff ends a wonderful review, which puts Teichmüller spaces into their historical context, as follows:-

The complex analytic structure is discussed(for the first time in over twenty years). In fact the whole structure of the basic theory is laid out for the first time. I really like this book. I think it is a wonderful place for someone who likes complex analysis to start learning the central ideas of the subject. Additionally, there are plenty of interesting analytic problems posed within the text. The writing is a model of clarity - we have come to expect no less from Lehto.

Although Lehto had permanent positions in Helsinki, from 1948 to 1983 he spent a total of ten years as a visiting professor abroad at institutions in the United States, India, Israel, and Japan. He relates a nice story in [1] about one of his early trips. On an international visit he was making in 1949 he was presented with a tea bag while on the flight. He had never seen a tea bag before and wondered whether to make a hole in it! In fact [1] contains fascinating information about Lehto's trips abroad when he met with leading mathematicians - he relates where he was sitting and what was eaten at these meetings.

There are other important contributions that Lehto has made to mathematics. One of these is his outstanding service to the International Mathematical Union. He organised the International Congress of Mathematicians held in Helsinki in August 1978. Following this, he was secretary of the International Mathematical Union from 1983 to 1988. In particular, he was Secretary when the Warsaw Congress, which was plagued with political problems, was held in 1983.

Lehto's speech at the closing session of the Warsaw Congress is at THIS LINK.

He spoke about the advantages of taking on that role [4]:-

My role as Secretary for the International Mathematical Union provided me with a brilliant view on the international world of mathematics. It opened up a whole network of new contacts and possibilities for me.

It also led to him publishing *Mathematics Without Borders: A History of the International Mathematical Union* (1998). The publisher writes:-

*The history of international mathematical co-operation over the last hundred years - from the first international congress in 1897 to plans for the World Mathematical Year 2000 - is a surprisingly compelling story. For reflected in the history of the International Mathematical Union is all the strife among world powers, as well as aspirations for cooperation among nations in an increasingly interdependent world. Yet throughout, the International Mathematical Union has sponsored international congresses around the world, and Professor Lehto's gripping story is one of individuals, among them many of the great mathematicians of our century, united in the common purpose of advancing their science, told against the backdrop of world events*.

...

Lehto has written other important historical works such as: his autobiography *Ei yliopiston voittanutta* Ⓣ (1999); a biography of Rolf Nevanlinna, *Korkeat maailmat* Ⓣ (2001), translated into German as *Erhabene Welten: Das Leben Rolf Nevanlinnas* Ⓣ (2008); a book about the brothers Vilho, Yrjö and Kalle Väisälä, *Oman tiensä kulkijat* Ⓣ (2004); and the book *Tieteen aatelia* Ⓣ (2008) about Ernst Lindelöf and his father, Lorenz Lindelöf, the founder of the Finnish Mathematical Society. Let us note that Lehto served as Chairman of the Finnish Mathematical Society from 1962 to 1985.

Throughout his career, Lehto has received honours and prizes which recognise his outstanding contributions. He has received honorary degrees from several universities, the Finnish Cultural Foundation Merit Award (1979), the Finnish Academy of Science Merit Award (1996), and he was one of eight people to receive a Public Information Lifetime Achievement Award from the Finnish Ministry of Education in 2009. The citation for this last mentioned award reads:-

The lifetime achievement award was presented to Academician Olli Lehto for his wide-ranging lifetime work in the fields of scientific research and public information. Lehto is one of the most internationally renowned mathematicians in Finland and an active writer on the topic of the history of science.

Finally, let us return to something we mentioned near the beginning of this biography, namely Lehto's love of butterflies. In fact [1] contains many details of how Lehto searched out butterflies in all the many locations he visited world-wide. Also he has published a book [2] which explains just how much the love of butterflies has dominated his life. We end with a quotation from [1] relating to his love of butterflies:-

A human artist has the potential to create beautiful shapes and colours, but nature never makes a mistake. When God colours and decorates the butterfly's wings, without exception the result appears to a human eye as extremely sophisticated and stylish. What supernatural power is it that controls this artistic design?

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