**Heisuke Hironaka**'s father ran a textile factory. This was not an occupation he wanted but he was forced into the family business when his father (Heisuke's paternal grandfather) died while he was still a schoolboy. Heisuke's mother was his father's third wife. After his first wife died, he remarried and had four children. However, his second wife then died and he married for the third time, to a younger sister of his second wife who, by this time, was widowed with one child. These five children comprised three boys and two girls. Heisuke's parents then had ten children of their own (six boys and four girls), so Heisuke grew up in a family of fifteen children. Of his parents' ten children, Heisuke was the second eldest having one older sister.

Two of Heisuke's half-brothers were killed, one fighting against the Americans and one fighting against the Chinese. Heisuke's father was devastated and as a consequence sold his textile factory and gave up work. Heisuke attended elementary school and then middle school where he began to develop a liking for mathematics. The town of Yamaguchi is about 80 km from Hiroshima and on Monday, 6 August 1945, at 8.15 in the morning, Heisuke's father witnessed the dropping of the atomic bomb on Hiroshima. They were fortunate that their area was not affected by the radiation. After middle school, Heisuke attended the boys' junior high school in Yanai. This meant that he had to take a 30 minute train journey to school every day. He started to learn to play the piano and became very keen but was advised by his teachers not to think of becoming a professional musician. After a mathematics professor from Hiroshima University gave a lecture at the junior high school, Heisuke became enthusiastic and applied to study mathematics at Hiroshima University. However he did not study for the entrance examination and failed. In the following year he applied to study physics at Kyoto University and was accepted. He was able to live with one of his sisters who had married and was living in Kyoto.

Kyoto University was founded in 1897 to train small numbers of selected students as academics. By the time Hironaka entered Kyoto University in 1949, it had been integrated into a mass higher education system but had maintained its prestige. In his first year Hironaka studied physics, chemistry and biology. In his second year, however, he began to realise that he was best suited to mathematics. By his third year as an undergraduate he had moved completely to courses in mathematics. Yasuo Akizuki, a pioneer of modern algebra in Japan, was a major influence on Hironaka during his time at Kyoto. He received his Bachelor of Science in 1954 and his Master of Science in 1956, both from Kyoto University. An important event happened in 1956 when Oscar Zariski visited Kyoto University [3]:-

When Zariski visited, I tried to tell him what I was doing. I have never been good in English! But my colleagues and teachers helped me to explain to him what I was doing. At some point Zariski said, "Maybe you can come to Harvard and study." And I said, "Okay."

Hironaka went to the United States in the summer of 1957 where he continued his studies at Harvard. He undertook research for his doctorate, with Oscar Zariski as his thesis advisor. While at Harvard, Hironaka became friends with Alexander Grothendieck who spent the academic year 1958-59 there. He invited Hironaka to the Institut des Hautes Etudes Scientifique in Paris in 1959-60 where he found himself the only visiting fellow. He was awarded a Ph.D. from Harvard in 1960 for his thesis *On the Theory of Birational Blowing-up*. He had already published three papers before submitting his thesis, *On the arithmetic genera and the effective genera of algebraic curves* (1957), *A note on algebraic geometry over ground rings. The invariance of Hilbert characteristic functions under the specialization process* (1958), and *A generalized theorem of Krull-Seidenberg on parameterized algebras of finite type* (1960). After completing his studies at Harvard, Hironaka was appointed to the staff at Brandeis University. Also in 1960 he married Wakako who was a Wien International Scholar at Brandeis University during 1958-60. She obtained an M.A. in Anthropology from Brandeis University Graduate School in 1964. Heisuke and Wakako Hironaka had one son Jo and one daughter Eriko. Wakako wrote several books, essays, translations, and critiques on education, culture, society, and women's issues. She entered Japanese politics being first elected to the House of Councillors in 1986. She has held high positions in the Democratic Party of Japan and in the Hosokawa Cabinet.

After being on the faculty at Brandeis University, Hironaka was appointed to Colombia University, and then to Harvard in 1968. In 1970 he had the distinction of being awarded a Fields Medal at the International Congress at Nice. This was for his work on algebraic varieties which we describe below.

Two algebraic varieties are said to be equivalent if there is a one-to-one correspondence between them with both the map and its inverse regular. Two varieties *U* and *V* are said to be birationally equivalent if they contain open sets *U'* and *V'* that are in biregular correspondence. Classical algebraic geometry studies properties of varieties which are invariant under birational transformations. Difficulties that arise as a result of the presence of singularities are avoided by using birational correspondences instead of biregular ones. The main problem in this area is to find a nonsingular algebraic variety *U*, that is birationally equivalent to an irreducible algebraic variety *V*, such that the mapping *f* : *U* → *V* is regular but not biregular.

Hironaka gave a general solution of this problem in any dimension in 1964 in Resolution of singularities of an algebraic variety over a field of characteristic zero. His work generalised that of Zariski who had proved the theorem concerning the resolution of singularities on an algebraic variety for dimension not exceeding 3. Jackson writes [3]:-

Taking a strikingly original approach, Hironaka created new algebraic tools and adapted existing ones suited to the problem. These tools have proved useful for attacking many other problems quite far removed from the resolution of singularities.

Hironaka talked about his solution in his lecture *On resolution of singularities (characteristic zero)* to the International Congress of Mathematicians in Stockholm in 1962.

In 1975 Hironaka returned to Japan where he was appointed a professor in the Research Institute for Mathematical Sciences of Kyoto University. He gave a course on the theory of several complex variables in 1977 and his lecture notes were written up and published in the book Introduction to analytic spaces (Japanese). Ikuo Kimura writes:-

Some fundamental theorems in the theory of several complex variables and of the geometry of complex manifolds are proved in a simple but rigorous form. ... Throughout this book one recognizes again the importance of the preparation theorem, and one finds good introductions to the study of the advanced theory of Stein spaces, the works of A Douady, and the theory of the resolution of singularities, to which Hironaka has contributed deeply.

Hironaka was Director of the Research Institute in Kyoto from 1983 to 1985, retiring in 1991 when he was made Professor Emeritus. However, in1996 he became president of Yamaguchi University, being inaugurated on 16 May. He held this position until 2002. He then became Academic Director of the University of Creation, a private university in Takasaki, Gunma, Japan.

We should also mention two educational projects which Hironaka has established (see [4] for details). Jackson writes [3]:-

Hironaka has contributed much time and effort to encouraging young people interested in mathematics. In1980, he started a summer seminar for Japanese high school students and later added one for Japanese and American college students; the seminars ran for more than two decades under his direction and continue to this day. To support the seminars he established a philanthropic foundation in1984called the Japan Association for Mathematical Sciences. The association also provides fellowships for Japanese students to pursue doctoral studies abroad.

Among the many honours he has received, in addition to the Fields Medal, we mention the Japan Academy Award in 1970 and the Order of Culture from Japan in 1975. He has been elected to the Japan Academy, the American Academy of Arts and Sciences and academies in France, Russia, Korea and Spain.

On 4-5 May 2009 the Clay Mathematics Institute held its 2009 Clay Research Conference in Harvard Science Center. Hironaka was invited to give one of the featured lectures on recent research advances and he spoke on *Resolution of Singularities in Algebraic Geometry*. The abstract for his talk reads:-

I will present my way of proving resolution of singularities of an algebraic variety of any dimension over a field of any characteristic. There are some points of general interest, I hope, technically and conceptually more than just the end result. The resolution problem for all arithmetic varieties(meaning algebraic schemes of finite type over the ring of integers)is reduced to the question of how to extend the result from modulo p(^{m}to modulo p^{}m+1)after a resolution of singularities overQ. I want to discuss certain problems which arise in this approach.

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

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