Heisuke Hironaka

2.1. Childhood and family background

Heisuke Hironaka was born on April 9, 1931, in Yu Town, Kuga District, Yamaguchi Prefecture, Japan, which is now part of Iwakuni. He was the seventh of fifteen children from his parents' remarried union. His father ran a prosperous fabric and textile factory, providing a comfortable upbringing before World War II. During the war, Hironaka was mobilized as a student to work in weapons manufacturing at the Hikari Naval Arsenal, interrupting his schooling. The war also brought personal tragedy, as two of his elder brothers were killed in action, and his family's business declined significantly. His father subsequently became a traveling clothing merchant to support the family.

As a child, Hironaka was known by his mother as "why-why boy" due to his insatiable curiosity and constant questioning. If his mother couldn't answer his inquiries, she would take him to local doctors or Shinto priests for explanations. Although he initially aspired to be a pianist and composer, even teaching himself piano in high school, he eventually abandoned this dream after receiving harsh criticism for a performance. His interest in mathematics was sparked in junior high school when he effortlessly solved a factoring problem that troubled his older sister, an experience he later described as the moment he first found mathematics truly interesting. A visiting mathematics professor from Hiroshima University also gave a lecture at his junior high school, further inspiring him to pursue the subject.

2.2. Academic journey in Japan

Hironaka initially applied to the undergraduate program at Hiroshima University but was not accepted. The following year, in 1949, he gained admission to Kyoto University to study physics. During his first two years, he explored a range of subjects including physics, chemistry, and biology. In his third year as an undergraduate, he decided to specialize in mathematics. He received his Bachelor of Science degree from Kyoto University in 1954 and continued his studies there, earning his Master of Science degree in 1956.

During his graduate studies at Kyoto University, Hironaka joined a seminar group led by Professor Yasuo Akizuki. Akizuki became a profound influence on Hironaka's mathematical development. This informal group, known as the "Akizuki School," delved into cutting-edge research topics, including the resolution of singularities problem, which would later become the focus of Hironaka's Fields Medal-winning work. While at Kyoto, Hironaka initially felt discouraged by the seemingly superior academic papers of other students. However, an encounter with a female student who returned a handkerchief he had dropped, calling him "teacher," inspired him to write his first academic paper. Although this initial paper was met with criticism, his second paper received positive reviews, leading to his master's degree. In 1963, Hironaka earned a PhD from Kyoto University for his thesis titled "The resolution of singularities of an algebraic variety."

A notable anecdote from this period involves a lecture Hironaka gave at the Mathematical Society of Japan in 1963 about the resolution of singularities problem. He proposed studying the problem by introducing various constraints due to its complexity. However, mathematician Kiyoshi Oka rose and argued that, to solve the problem, one should not impose restrictions but rather idealize and tackle an even more difficult version. Hironaka later adopted this idealized approach, leading to his Fields Medal-winning work.

2.3. Doctoral studies abroad

In 1956, Professor Akizuki invited Oscar Zariski, a distinguished professor of mathematics from Harvard University, to Kyoto University. Hironaka seized this opportunity to present his own research to Zariski, who was deeply impressed and suggested that Hironaka relocate to Harvard University to continue his studies. In 1957, Hironaka moved to the United States and began his doctoral studies at Harvard under Zariski's supervision. His strong background in commutative algebra, cultivated under Akizuki's tutelage in Kyoto, provided him with unique insights that enriched the mathematics discussions at Harvard, which at the time had a greater emphasis on geometric perspectives.

During his time at Harvard, Hironaka also interacted with other influential mathematicians, including Alexander Grothendieck, who visited Harvard in 1958-1959. Grothendieck's visit proved to be another significant influence on Hironaka, and Grothendieck invited him to the Institut des Hautes Études Scientifiques (IHES) in Paris. Hironaka spent six months as a visiting researcher at IHES in 1959. During this period in Paris, he befriended conductor Seiji Ozawa at a language school. Their friendship continued when Ozawa visited the U.S. in 1960 for the Berkshire Music Festival (now Tanglewood Music Festival), with Hironaka driving to pick him up from the airport.

Upon returning to Harvard in 1960, Hironaka successfully completed his PhD, submitting his thesis titled "On the Theory of Birational Blowing-up." He studied alongside other notable algebraic geometers, including David Mumford, Michael Artin, and Steven Kleiman, with Mumford and Artin providing significant assistance during his Harvard years.

3.1. University professorships

After completing his doctoral studies, Hironaka began his teaching career as an Associate Professor of Mathematics at Brandeis University from 1960 to 1963, where he also served as an Assistant Professor from 1961 and an Associate Professor from 1963. He then moved to Columbia University, where he taught as a Professor of mathematics from 1964 to 1968. In 1968, he joined the faculty at Harvard University as a professor of mathematics, a position he held until 1992, when he became an emeritus professor, holding the William Elwood Byerly Professorship in Mathematics.

Concurrently, Hironaka maintained strong ties with Japanese academia. He returned to Japan for a joint professorship at the Research Institute for Mathematical Sciences (RIMS) and Kyoto University from 1975 to 1988, serving as the Director of RIMS from 1983 to 1985.

3.2. University leadership and visiting appointments

Beyond his professorial roles, Hironaka took on significant leadership positions in university administration. He served as the president of Yamaguchi University from 1996 to 2002. In 2004, he became the chairman of the Sozo Gakuen University (now defunct), a position he held until 2013.

Hironaka has also been active as a visiting professor internationally. In 2008, he was invited to Seoul National University in South Korea as an endowed professor (석좌교수_{Seokjwa Gyosu}^Korean), a position he held actively for many years.

4.1. Resolution of singularities theorem

In 1964, Hironaka published his landmark proof demonstrating that singularities of algebraic varieties can be "resolved" in characteristic zero. This means that any algebraic variety can be replaced by (or, more precisely, is birationally equivalent to) a similar variety that has no singularities. This fundamental result provided a general solution to a long-standing problem in algebraic geometry, which had roots in the work of mathematicians like Max Noether and Oscar Zariski. His proof was published in two parts in the *Annals of Mathematics*, a prestigious journal published by Princeton University.

Hironaka recounted that he felt very close to solving this problem during his studies at Harvard. Shortly after securing his first teaching position at Brandeis University, he realized that by combining his expertise in commutative algebra from Kyoto, the geometry of polynomials from Harvard, and the globalization techniques learned at IHES in Paris, he possessed all the necessary tools to solve the problem. His solution to the resolution of singularities in characteristic zero earned him the Fields Medal in 1970.

4.2. Other significant works

In 1960, Hironaka introduced a crucial concept known as Hironaka's example. This example demonstrated that a deformation of Kähler manifolds does not necessarily have to be Kähler. Specifically, it described a one-parameter family of smooth compact complex 3-manifolds where most fibers are Kähler (and even projective), but one specific fiber is not Kähler. This example is significant because it illustrates that several plausible statements, which hold true for smooth varieties of dimension at most 2, fail for smooth varieties of dimension 3 or higher.

More recently, in 2017, Hironaka posted a manuscript on his personal webpage claiming to provide a proof for the existence of a resolution of singularities in positive characteristic. This remains a highly significant and challenging open problem in algebraic geometry.

5.1. Mathematical philosophy and interests

Hironaka's approach to mathematics is deeply personal and intuitive. He famously described his interest in the resolution of singularities problem as possessing the logic and mystery akin to "a boy falling in love with a girl," highlighting the blend of intellectual curiosity and emotional engagement that drives his work. His overarching philosophy is encapsulated in his motto, "素心深考_{Soshin Shinkō}^Japanese," which emphasizes a pure heart and deep contemplation.

He finds mathematical interest in everyday phenomena, stating, "I accumulate anything to do with numbers. For instance, I have more than 10,000 photos of flowers and leaves. I like to just count the numbers and compare them. I am so pleased to be a mathematician, because I can see the mathematical interest in things." This perspective underscores his belief in the pervasive nature of mathematics and his joy in discovering its principles in the mundane.

5.2. Promotion of mathematics education

Hironaka has been a passionate advocate for mathematics education, particularly in Japan and South Korea, dedicating significant efforts to popularize the subject and nurture young talent.

In 1980, he initiated a summer seminar for Japanese high school students. This program later expanded to include college students from both Japan and the United States, fostering cross-cultural academic exchange. To provide stable funding for these seminars and related initiatives, Hironaka established the Japanese Association for Mathematical Sciences (JAMS, 数理科学振興会_{Sūri Kagaku Shinkōkai}^Japanese) in 1984, where he served as executive director. JAMS received additional support from both corporations and the Japanese government. Shing-Tung Yau, a prominent Harvard mathematics professor emeritus, has acknowledged the enduring impact of Hironaka's efforts, noting that many former JAMS fellows have gone on to become mathematics professors across the United States and Japan, a testament to the program's success in an era with limited international grant opportunities.

Beyond JAMS, Hironaka's contributions to education include serving as the chairman of the "Arithmetic Olympics" (算数オリンピック_{Sansū Orinpikku}^Japanese) starting in 1992, an event designed to promote interest in mathematics among elementary school students. He also supervised mathematics textbooks published by Tokyo Shoseki.

His influence extended to South Korea during his tenure as a visiting professor at Seoul National University from 2008 to 2009. There, he notably mentored an undergraduate student named June Huh. Huh, who had previously dropped out of high school and aspired to be a poet, found inspiration in Hironaka's lectures. Hironaka encouraged Huh's interest in pursuing mathematics for graduate school, leading to a close mentorship where Huh even visited Hironaka's home in Kyoto. Huh later went on to win the Fields Medal in 2022 for his work connecting algebraic geometry and combinatorics, an achievement widely attributed to Hironaka's formative guidance and encouragement.

8.1. Authored books and papers

• "On the arithmetic genera and the effective genera of algebraic curves," Mem. College Sci. Univ. Kyoto Ser. A Math., 30(2): 177-195 (1957). [https://projecteuclid.org/journals/memoirs-of-the-college-of-science-university-of-kyoto-series-a-mathematics/volume-30/issue-2/On-the-arithmetic-genera-and-the-effective-genera-of-algebraic/10.1215/kjm/1250777055.full Project Euclid: On the arithmetic genera and the effective genera of algebraic curves]
• "On the theory of birational blowing-up" (1960). [https://id.lib.harvard.edu/alma/990038590670203941/catalog Harvard Library: On the theory of birational blowing-up]
• "An example of a non-Kählerian complex-analytic deformation of Kählerian complex structures," Annals of Mathematics, 75(1): 190-208 (1962).
• "The Resolution of Singularities of an Algebraic Variety over a Field of Characteristic Zero," Annals of Mathematics, 79(1): 109-203 (1964). [https://www.jstor.org/stable/1970486 JSTOR: The Resolution of Singularities of an Algebraic Variety over a Field of Characteristic Zero]
• "On the characters $\backslash nu$ and $\backslash tau$ of singularities," J. Math. Kyoto Univ., 7(1): 325-327 (1967). [https://projecteuclid.org/journals/kyoto-journal-of-mathematics/volume-7/issue-1/On-the-characters-nu-and-tau-of-singularities/10.1215/kjm/1250524306.full Project Euclid: On the characters ν and τ of singularities]
• "The theory of infinitely near singular points," J. Korean Math. Soc. 40(5): 901-920 (1974). [https://koreascience.kr/article/JAKO200311921892201.page KoreaScience: The theory of infinitely near singular points]
• The Possibility to Pull Out the Ability (広中平祐の家庭教育論 可能性をひきだす教育), Kodansha (1978).
• Hironaka Heisuke's Education Exploration (広中平祐の教育探検), NHK Publishing (1979).
• Creating with Variable Thinking: A Text for Ideas in Planning, Education, and Technology (「可変思考」で創造しよう 企画・教育・技術への発想テキスト), Kobunsha (1982).
• The Discovery of Academia (学問の発見), Kosei Publishing (1982), later reissued as To Live and To Learn (生きること学ぶこと) by Shueisha Bunko (1984).
• Science's Wisdom, Heart's Wisdom (科学の知恵心の智慧), Kosei Publishing (1985).
• Like Clouds (雲の如く), Obunsha (1986).
• Theory of a Springhead Nation: What to Create After Prosperity (湧源国家論 豊かさの後に何を創造するのか), PHP Institute (1988).
• "Fame, sweet and bitter." In P. Hilton, F. Hirzebruch, and R. Remmert (Eds.), Miscellanea mathematica (pp. 155-176) (1991). [https://link.springer.com/chapter/10.1007/978-3-642-76709-8_10 Springer: Fame, sweet and bitter]
• Algebraic Geometry (代数幾何学), lectures recorded by Shigefumi Mori, edited by Masaki Maruyama, Jun Moriaki, and Shu Kawaguchi, Kyoto University Press (2004).

8.2. Collaborative and edited works

• Mathematics and Eroticism (数学とエロチシズム), co-authored with Masuo Ikeda, Kodansha (1977).
• Speaking of Japan (日本を語る), co-authored with Leo Esaki, Mainichi Shimbunsha (1977).
• Having a Soft Heart: Our Luck, Dullness, and Root (やわらかな心をもつ ぼくたちふたりの運・鈍・根), dialogue with Seiji Ozawa, Soseiki (1977).
• Children's Education and Parental Involvement (子供の教育と親のかかわり), co-authored with Wakako Hironaka, Seikyo Shimbunsha (1979).
• Youth's Wings: 12 Chapters for Young Japanese (青年の翼 若い日本人のための12章), edited, Pan-Asian (1980).
• Introduction to Analytic Space (解析空間入門), co-authored with Tosuke Urabe, Asakura Shoten (1981).
• Hironaka Heisuke and Wakako's Educational Consultation: For Developing Intellectual Desire and Individuality (広中平祐・和歌子の教育相談 知欲と個性を伸ばすために), co-authored with Wakako Hironaka, Gakken (1982).
• The Discovery of Wisdom: Wisdom (知の発見-叡知), co-authored with Reifu Fujisawa, Terakoya Publishing (1983).
• Jellyfish Mathematics, Si Alien Chemistry (クラゲの数学・Si星人の化学), co-authored with Chuji Tsuboi and Hiroshi Ezawa, Baifukan (1985).
• Parents, Let's Think Now: Aiming for an "Environment" that Develops Children's Academic Ability and Individuality (親ならいまこそ考えよう 子供の学力、個性を伸ばす"環境"づくりをめざして... 対談), co-authored with Kinpei Namba, JCA Publishing Bureau (1986).
• Living with a Pure Heart and Wish: Dialogue (素心・素願に生きる 対話), dialogue with Tsutomu Mizukami, Shogakukan (1989).
• Encyclopedia of Contemporary Mathematical Sciences (現代数理科学事典), edited, Osaka Shoseki (1991).
• Life from Sixty: Living by Fate, Giving Guidance, Courage, Hope (人生は六十歳から 天命に生きる 生きる指針・勇気・希望を与える), co-authored with Kazuo Wada, Diamond Sales Editorial Planning (1994).
• "Formal functions and formal imbeddings" / by Heisuke Hironaka and Hideyuki Matsumura (1967).
• "The theory of the maximal contact" / José M. Aroca, Heisuke Hironaka and José L. Vicente (1975).
• "Desingularization theorems" / Jose M. Aroca, Heisuke Hironaka and Jose L. Vicente (1977).
• "Geometric singularity theory" / editors of the volume, Heisuke Hironaka, Stanisław Janeczko (2004).
• "Complex Analytic Desingularization" / Aroca, J. M.; Hironaka, H; Vicente, J. L. (2018).

8.3. Translated works

• Supervised the translation of Benoit Mandelbrot's The Fractal Geometry of Nature (フラクタル幾何学), Nikkei Science (1985), later Chikuma Gakugei Bunko (2011). The translation work itself was carried out by the Yugen Club, associated with JAMS.