Shigefumi Mori

Shigefumi Mori
Shigefumi Mori
Born	森 重文 23 2, 1951
Birthplace	Nagoya, Japan
Nationality	Japanese
Occupation	Mathematician
Employer	Kyoto University
Known for	Algebraic geometry, minimal model program, Keel–Mori theorem
Education	Kyoto University (Ph.D., 1978)
Awards	Fields Medal (1990), Cole Prize (1990)

Shigefumi Mori (森 重文, Mori Shigefumi; born February 23, 1951) is a Japanese mathematician whose work in algebraic geometry transformed the study of higher-dimensional varieties and earned him the highest honor in mathematics. Born in Nagoya, Japan, Mori developed the minimal model program (also known as Mori's program), a sweeping framework for classifying algebraic varieties of dimension three and higher that extended the classical theory of surfaces into new mathematical territory. For this achievement, he was awarded the Fields Medal in 1990, becoming one of only a handful of Japanese mathematicians to receive the prize. In addition to the Fields Medal, he received the Frank Nelson Cole Prize in Algebra from the American Mathematical Society in the same year. Mori spent much of his career at Nagoya University and Kyoto University, where he was affiliated with the Research Institute for Mathematical Sciences (RIMS). In 2014, he was elected President of the International Mathematical Union (IMU), serving from 2015 to 2018, a role that placed him at the center of global efforts to promote mathematical research and collaboration.

Early Life

Shigefumi Mori was born on February 23, 1951, in Nagoya, a major city in central Japan's Aichi Prefecture. Growing up in postwar Japan during a period of rapid economic and educational development, Mori developed an early interest in mathematics. Nagoya, home to one of Japan's leading research universities, provided an intellectual environment that fostered academic pursuits. Details of Mori's childhood and family background are not extensively documented in public sources, but his trajectory from Nagoya to Kyoto University indicates an early aptitude for rigorous mathematical thinking.

Mori's formative years coincided with a period when Japanese mathematics was gaining increasing international recognition. The legacy of earlier Japanese mathematicians, including members of the Kyoto school of algebraic geometry, provided a rich intellectual tradition into which the young Mori would eventually enter. His decision to pursue mathematics at Kyoto University placed him at one of the foremost centers of mathematical research in Japan.

Education

Mori pursued his undergraduate and graduate studies at Kyoto University, one of Japan's most prestigious institutions for mathematical research. He completed his doctoral thesis, titled "The Endomorphism Rings of Some Abelian Varieties," in 1978 under the supervision of Masayoshi Nagata, a distinguished algebraic geometer known for his contributions to commutative algebra and his resolution of Hilbert's fourteenth problem. Nagata's influence on Mori was significant; studying under one of Japan's foremost algebraists provided Mori with a deep foundation in the techniques and problems of algebraic geometry. The thesis on abelian varieties dealt with fundamental algebraic structures, and while it addressed a classical topic, Mori's subsequent work would rapidly move into newer and more ambitious territory.

Career

Early Academic Work and the Hartshorne Conjecture

After completing his doctorate in 1978, Mori began his academic career in Japan's university system, initially holding positions at Kyoto University before moving to Nagoya University. His early research quickly attracted international attention. One of his first major results was his proof of the Hartshorne conjecture on projective spaces, which he achieved in 1979. The conjecture, posed by the American algebraic geometer Robin Hartshorne, concerned the characterization of projective spaces among smooth projective varieties. Mori's proof was remarkable not only for its result but also for the techniques he introduced, including his method of "bend and break," which involved studying rational curves on algebraic varieties.

The "bend and break" technique became one of Mori's signature contributions to algebraic geometry. The method demonstrates that under certain positivity conditions on the tangent bundle of a variety, one can produce rational curves — copies of the projective line — on the variety by degenerating families of curves until they "break" into components. This deformation-theoretic approach proved to be extraordinarily powerful and became a foundational tool in higher-dimensional algebraic geometry. The proof of the Hartshorne conjecture established Mori, still in his late twenties, as a mathematician of exceptional ability.

The Minimal Model Program

Mori's most consequential contribution to mathematics is the development and advancement of the minimal model program (MMP), sometimes referred to as "Mori's program" or "Mori theory." The minimal model program is a systematic framework for classifying algebraic varieties — the central objects of study in algebraic geometry — by simplifying them to canonical or "minimal" forms through a sequence of birational transformations.

The historical context for this program lies in the classical theory of algebraic surfaces, developed in the late nineteenth and early twentieth centuries by Italian and other European mathematicians. For surfaces (two-dimensional varieties), mathematicians had established that every smooth projective surface could be reduced to a minimal model through a finite sequence of "blowing down" operations that contract certain special curves (exceptional curves of the first kind). This minimal model is then either a surface with nonnegative Kodaira dimension — a numerical invariant that measures the "complexity" of the variety — or a ruled surface, which is fibered over a curve with rational fibers.

Extending this classification to varieties of dimension three (three-folds) and higher was one of the outstanding problems in algebraic geometry during the latter half of the twentieth century. The difficulty lay in the fact that the higher-dimensional analogue of the contraction process can introduce singularities that do not arise in the surface case, and new types of birational operations — specifically, "flips" — are needed to continue the simplification process. Mori's program provided both the conceptual framework and specific technical results needed to carry out this generalization.

In a series of foundational papers during the 1980s, Mori established the existence of the necessary birational contractions for three-folds, classified the types of contractions (divisorial contractions and flipping contractions), and proved the existence of flips in dimension three. The proof of the existence of three-dimensional flips, published in 1988, was a tour de force of algebraic geometry. This result, combined with his earlier work, showed that the minimal model program could be carried out for three-folds: every smooth projective three-fold can be transformed, through a finite sequence of divisorial contractions and flips, into either a minimal model (with nef canonical divisor) or a Mori fiber space (a variety fibered over a lower-dimensional base with Fano fibers).

The significance of this achievement cannot be overstated in the context of algebraic geometry. It represented the first successful extension of the classification theory beyond dimension two and opened the door to understanding the birational geometry of higher-dimensional varieties. The techniques Mori introduced — particularly the study of extremal rays in the cone of curves, the contraction theorem, and the cone theorem — became standard tools in the field.

Influence on Higher-Dimensional Geometry

Mori's work laid the groundwork for subsequent developments in birational geometry that extended the minimal model program to dimensions greater than three. In 2006, Caucher Birkar, Paolo Cascini, Christopher D. Hacon, and James McKernan proved the existence of flips in all dimensions, a result that built directly on the foundations established by Mori's program. This breakthrough, for which the four mathematicians received the AMS E. H. Moore Research Article Prize in 2016, was described as resolving a central conjecture in the minimal model program. McKernan subsequently received the 2018 Breakthrough Prize in Mathematics for "transformational contributions to birational algebraic geometry, especially to the minimal model program in all dimensions."

The Keel–Mori theorem, a result obtained jointly with Seán Keel, is another notable contribution. This theorem addresses the existence of coarse moduli spaces for algebraic stacks satisfying certain conditions, providing an important bridge between the abstract theory of stacks and more concrete geometric objects. The result has found applications throughout algebraic geometry and moduli theory.

Positions at Nagoya and Kyoto Universities

Throughout his career, Mori held academic positions at two of Japan's leading research universities. He was a professor at Nagoya University, one of the original Imperial Universities and a center for mathematical research, before moving to Kyoto University, where he joined the Research Institute for Mathematical Sciences (RIMS). RIMS, founded in 1963, is one of the world's foremost institutes dedicated to mathematical research, and Mori's appointment there placed him alongside other leading Japanese mathematicians. He served as director of RIMS and became one of its most prominent members.

Mori also spent time at research institutions outside Japan. He was a visiting scholar at Harvard University and the Institute for Advanced Study in Princeton, among other institutions, which facilitated international collaborations and broadened the impact of his work.

Presidency of the International Mathematical Union

In August 2014, Mori was elected President of the International Mathematical Union (IMU), the premier international organization for mathematical research and cooperation. He assumed the presidency in January 2015 and served through 2018. The IMU is responsible for organizing the International Congress of Mathematicians (ICM), held every four years, and for awarding the Fields Medal, the Nevanlinna Prize (now the IMU Abacus Medal), and other major prizes in mathematics.

As IMU President, Mori presided over the 2018 International Congress of Mathematicians in Rio de Janeiro, Brazil. His term also coincided with growing international discussions about diversity and inclusion in mathematics, including efforts to address the gender gap in the discipline. Mori's presidency placed a Japanese mathematician at the helm of the global mathematical community, reflecting both his personal standing and the strength of Japanese mathematics on the international stage.

Continued Engagement

In more recent years, Mori has continued to participate in international mathematical events. In May 2015, he delivered a lecture at the University of Macau as part of their University Lecture Series, where he was introduced as a Fields Medal recipient and a figure of international standing in mathematics. He has remained active in the broader mathematical community through lectures, advisory roles, and contributions to mathematical organizations.

Personal Life

Shigefumi Mori has maintained a relatively private personal life throughout his career. He has been based primarily in the Kyoto and Nagoya areas of Japan for much of his professional life, in keeping with his academic appointments at Nagoya University and Kyoto University's Research Institute for Mathematical Sciences. Public records do not extensively document his family life, and he has not been a frequent subject of popular media profiles outside the context of his mathematical achievements. In 2021, he was mentioned in Japanese media coverage related to cultural and academic affairs in Japan.

Recognition

Mori's contributions to algebraic geometry have been recognized with several of the highest honors in mathematics. His two most significant awards, both received in 1990, are the Fields Medal and the Frank Nelson Cole Prize in Algebra.

The Fields Medal, awarded by the International Mathematical Union at the International Congress of Mathematicians, is often described as the most prestigious award in mathematics. It is given every four years to mathematicians under the age of forty who have made outstanding contributions. Mori received the Fields Medal at the 1990 ICM in Kyoto, Japan, for his work on the classification of three-folds and the minimal model program. The citation recognized his proof of the Hartshorne conjecture and his development of the theory of extremal rays, which provided the framework for the birational classification of higher-dimensional algebraic varieties. Mori was 39 years old at the time of the award.

The Frank Nelson Cole Prize in Algebra, awarded by the American Mathematical Society, recognized the same body of work. The Cole Prize is one of the oldest prizes awarded by the AMS and is given for outstanding contributions to algebra.

Mori also received recognition from the Japanese government and academic institutions. His election as President of the International Mathematical Union in 2014 served as further international recognition of his stature in the mathematical community.

His work is cited extensively in the mathematical literature. Author profiles in the Mathematical Reviews (MathSciNet) database and the zbMATH database document the scope of his publications and their influence on subsequent research.

Legacy

Shigefumi Mori's work fundamentally reshaped the field of algebraic geometry. The minimal model program, which bears his name informally as "Mori's program," became one of the central organizing frameworks of the discipline in the late twentieth and early twenty-first centuries. By demonstrating that the classical birational classification of surfaces could be extended to three-folds — and by providing the tools and techniques that would eventually allow extension to all dimensions — Mori opened a new chapter in algebraic geometry.

The specific technical innovations he introduced have become standard in the field. The cone theorem, the contraction theorem, the theory of extremal rays, and the bend-and-break technique are taught in graduate courses on algebraic geometry worldwide and appear in standard textbooks on the subject. His proof of the existence of flips in dimension three provided the key missing ingredient for the three-dimensional minimal model program and set the stage for the work of Birkar, Cascini, Hacon, and McKernan, who extended these results to arbitrary dimensions.

As a leader of the International Mathematical Union, Mori also contributed to the institutional and organizational dimensions of mathematics. His presidency of the IMU from 2015 to 2018 placed him in a position to shape the direction of international mathematical cooperation, including the organization of the International Congress of Mathematicians and the selection of prize laureates.

Mori's career represents a continuation of the strong Japanese tradition in algebraic geometry, building on the work of predecessors such as his doctoral advisor Masayoshi Nagata, as well as Kunihiko Kodaira, who received the Fields Medal in 1954 for his work on algebraic and complex geometry. Together with his contemporaries, Mori helped establish Japan as one of the leading centers for research in algebraic geometry.

The volume Mathematics: Frontiers and Perspectives, published by the American Mathematical Society and edited by V. I. Arnold, Michael Atiyah, Peter D. Lax, and Barry Mazur, included contributions reflecting the state of mathematics at the turn of the millennium, a period in which Mori's program was one of the central achievements under discussion.