William Thurston

William Thurston
Thurston in 1991
William Thurston
Born	William Paul Thurston 30 10, 1946
Birthplace	Washington, D.C., U.S.
Died	Template:Death date and age Rochester, New York, U.S.
Nationality	American
Occupation	Mathematician
Known for	Geometrization conjecture, low-dimensional topology, foliation theory
Education	Ph.D., University of California, Berkeley (1972)
Awards	Fields Medal (1982), Oswald Veblen Prize in Geometry (1976), Leroy P. Steele Prize (2012)

William Paul Thurston (October 30, 1946 – August 21, 2012) was an American mathematician whose work fundamentally reshaped the study of low-dimensional topology and geometry. Born in Washington, D.C., Thurston spent a career that spanned appointments at some of the most prominent mathematical institutions in the United States, including Princeton University, the University of California, Davis, and Cornell University, where he held the title of Jacob Gould Schurman Professor of Mathematics at the time of his death. He was awarded the Fields Medal in 1982 for his contributions to the study of three-manifolds, and received numerous other distinctions throughout his life, including the Oswald Veblen Prize in Geometry in 1976, the Alan T. Waterman Award in 1979, and the Leroy P. Steele Prize in 2012. Thurston also served as director of the Mathematical Sciences Research Institute. His geometrization conjecture, which proposed that every closed three-manifold could be decomposed into pieces admitting one of eight geometric structures, became one of the most important guiding problems in topology and was ultimately proved by Grigori Perelman in 2003 as part of the resolution of the Poincaré conjecture. Thurston died on August 21, 2012, in Rochester, New York, at the age of 65.

Early Life

William Paul Thurston was born on October 30, 1946, in Washington, D.C. He grew up in a period of expanding American scientific ambition and showed mathematical aptitude from an early age. Details of his childhood and family background are not extensively documented in public sources, but his early intellectual development led him to pursue undergraduate studies at New College of Florida, a small liberal arts institution in Sarasota known for its emphasis on independent study and academic rigor.

Thurston's time at New College provided a foundation for the highly creative and visual approach to mathematics that would later distinguish his career. The college's tutorial-based curriculum, which emphasized self-directed study and close faculty interaction rather than traditional grading, may have nurtured the kind of independent thinking that characterized Thurston's later mathematical style. He completed his undergraduate education at New College before proceeding to graduate work at the University of California, Berkeley, one of the leading centers for mathematical research in the United States.

Education

Thurston enrolled in the doctoral program in mathematics at the University of California, Berkeley, where he studied under the supervision of Morris Hirsch, a prominent figure in differential topology. His doctoral thesis, titled "Foliations of three-manifolds which are circle bundles," was completed in 1972. The thesis addressed fundamental questions about foliations — decompositions of manifolds into collections of lower-dimensional submanifolds — and marked the beginning of Thurston's deep engagement with the topology and geometry of three-dimensional spaces. The work demonstrated an exceptional capacity for geometric intuition that would become a hallmark of his subsequent contributions.

Berkeley in the early 1970s was a fertile environment for topology and geometry, and Thurston's graduate years placed him at the intersection of several active research programs. His doctoral work on foliations quickly attracted attention from the broader mathematical community and established him as a rising figure in the field even before he completed his degree.

Career

Early Work on Foliations

Following the completion of his doctorate in 1972, Thurston began producing results on foliation theory at a remarkable pace. His early work addressed the existence and classification of foliations on manifolds, and he proved several foundational results in the area. Among his most striking early achievements was demonstrating that any compact manifold with Euler characteristic zero admits a codimension-one foliation. These results transformed the understanding of foliations and established Thurston as one of the leading young mathematicians in the world.

The significance of Thurston's early contributions was recognized in 1976, when he received the Oswald Veblen Prize in Geometry from the American Mathematical Society, one of the most prestigious awards in the field of geometry and topology. This was followed in 1979 by the Alan T. Waterman Award from the National Science Foundation, given annually to an outstanding young researcher in any field of science or engineering. These honors reflected the mathematical community's recognition that Thurston's work had opened new directions in topology.

Princeton University and the Geometrization Conjecture

Thurston joined the faculty of Princeton University, where he held a professorship in the Department of Mathematics. It was during his time at Princeton that he developed the ideas that would become his most celebrated contribution: the geometrization conjecture for three-manifolds.

The geometrization conjecture, formulated in the late 1970s, proposed that every closed three-dimensional manifold can be cut along certain surfaces into pieces, each of which admits one of eight types of geometric structure. These eight model geometries — which include Euclidean, hyperbolic, and spherical geometry, among others — provided a complete framework for understanding the geometry of three-manifolds. The conjecture represented a far-reaching generalization of the uniformization theorem for surfaces, which states that every closed two-dimensional surface admits a geometry of constant curvature.

Central to Thurston's program was the assertion that hyperbolic geometry plays a dominant role in the world of three-manifolds — that, in a precise sense, "most" three-manifolds are hyperbolic. This insight was supported by Thurston's proof of the hyperbolization theorem for Haken manifolds, a large and important class of three-manifolds. The proof demonstrated that Haken manifolds satisfying certain topological conditions admit hyperbolic structures, and it represented one of the deepest results in three-dimensional topology at the time.

Thurston's approach was notable for its geometric imagination. Rather than relying primarily on algebraic or combinatorial methods, he developed an intensely visual and intuitive understanding of three-dimensional spaces. His lectures and writings conveyed a sense that geometric objects could be seen and manipulated in the mind's eye, and this perspective influenced a generation of mathematicians who followed his work.

Fields Medal

In 1982, Thurston was awarded the Fields Medal at the International Congress of Mathematicians. The Fields Medal, often described as the highest honor in mathematics, is awarded every four years to mathematicians under the age of 40 who have made outstanding contributions. Thurston received the medal for his work on the topology and geometry of three-manifolds, including the geometrization conjecture and the hyperbolization theorem for Haken manifolds.

The Fields Medal citation recognized Thurston's transformative impact on the field. His work had established new connections between topology, geometry, and group theory, and had provided a unifying vision for the study of three-manifolds that would guide research for decades. Thurston was elected to the National Academy of Sciences in 1983, the year following his Fields Medal.

Mathematical Sciences Research Institute

Thurston served as director of the Mathematical Sciences Research Institute (MSRI) in Berkeley, California. MSRI is one of the premier mathematical research institutes in the world, hosting programs and workshops that bring together leading researchers across all areas of mathematics. In this role, Thurston was responsible for overseeing the institute's scientific programs and fostering collaboration among mathematicians from diverse subfields.

His tenure as director reflected his commitment to the broader mathematical community and his interest in mathematical communication and education, themes that became increasingly prominent in his later career.

University of California, Davis

Following his period of leadership at MSRI, Thurston joined the faculty of the University of California, Davis, where he served as a professor of mathematics. His appointment at UC Davis continued his pattern of engagement with both research and education at major American universities.

Cornell University

Thurston subsequently moved to Cornell University, where he held the Jacob Gould Schurman Professorship of Mathematics, a distinguished endowed chair. He remained at Cornell for the remainder of his career. At Cornell, he continued his research activities and mentored graduate students. According to the Mathematics Genealogy Project, Thurston supervised a substantial number of doctoral students over the course of his career, many of whom went on to make significant contributions to mathematics in their own right.

Influence on Mathematical Communication

Beyond his technical contributions, Thurston was known for his reflections on the nature of mathematical understanding and communication. He articulated views about how mathematical knowledge is transmitted and how the social structure of the mathematical community affects the development of ideas. His 1994 essay "On Proof and Progress in Mathematics," published in the Bulletin of the American Mathematical Society, addressed the question of what constitutes mathematical understanding and argued that proof is only one component of mathematical progress, with intuition, communication, and shared understanding playing equally important roles.

This perspective was sometimes the subject of discussion within the mathematical community. A 2012 article in Scientific American discussed Thurston's influence on debates about the role of proof in mathematics, noting that his approach to mathematical communication had raised important questions about how mathematical knowledge is validated and transmitted.

Published Works

Thurston's mathematical writings include the influential book Three-Dimensional Geometry and Topology, published by Princeton University Press. The book grew out of lecture notes that had circulated widely in the mathematical community for years before formal publication and served as a key reference for the geometrization program. The work received the Doob Prize from the American Mathematical Society in 2005, an award recognizing an outstanding mathematical book.

Thurston's lecture notes on the geometry and topology of three-manifolds, sometimes referred to as the "Thurston notes," were widely distributed in unpublished form and had a significant impact on the field even before their partial incorporation into the published book. These notes introduced many mathematicians to the ideas of the geometrization program and served as a foundation for further research.

Personal Life

Thurston died on August 21, 2012, in Rochester, New York, at the age of 65. The New York Times published an obituary noting his death and his contributions to mathematics. Cornell University's announcement described him as a "world-renowned mathematician" and the Jacob Gould Schurman Professor of Mathematics at the university.

Thurston's death prompted tributes from mathematicians around the world. Scientific American published reflections on his mathematical legacy, noting the breadth and depth of his influence on the field.

Recognition

Thurston received numerous awards and honors over the course of his career, reflecting the significance of his contributions to mathematics:

• Oswald Veblen Prize in Geometry (1976) — Awarded by the American Mathematical Society for his work on foliations and the topology of manifolds.
• Alan T. Waterman Award (1979) — Given by the National Science Foundation to recognize an outstanding young American researcher.
• Fields Medal (1982) — Awarded at the International Congress of Mathematicians for his contributions to the study of three-manifolds.
• National Academy of Sciences (1983) — Elected as a member.
• Doob Prize (2005) — Awarded by the American Mathematical Society for his book Three-Dimensional Geometry and Topology.
• Leroy P. Steele Prize (2012) — Awarded by the American Mathematical Society for seminal contribution to research. Thurston received this prize posthumously or near the end of his life for his work on the geometrization conjecture.

These awards span the major categories of mathematical recognition and reflect both the depth of Thurston's individual contributions and his broader impact on the direction of mathematical research.

Legacy

William Thurston's legacy in mathematics rests primarily on his transformation of the study of three-dimensional manifolds. His geometrization conjecture provided a comprehensive framework for understanding the structure of three-manifolds, analogous to the classification of surfaces in two dimensions. The conjecture guided research in topology and geometry for more than two decades and was ultimately proved by Grigori Perelman in 2002–2003, using Richard Hamilton's Ricci flow program. Perelman's proof, which also resolved the century-old Poincaré conjecture as a special case, confirmed the correctness of Thurston's geometric vision.

The impact of Thurston's work extended well beyond the geometrization conjecture itself. His introduction of hyperbolic geometry as a central tool in three-dimensional topology opened new connections between topology, differential geometry, and geometric group theory. Concepts and techniques developed by Thurston — including the theory of hyperbolic structures on three-manifolds, the Thurston norm, and the theory of automatic groups — became standard tools in the field.

Thurston's influence was also felt through his mentorship of graduate students and his role in shaping mathematical culture. His emphasis on geometric intuition and visual understanding offered an alternative to more algebraic approaches and inspired many mathematicians to develop new ways of thinking about topological problems. The American Mathematical Society's memorial notices described the breadth of his influence and the originality of his mathematical vision.

The American Mathematical Society announced his death in 2012 with a statement reflecting on his contributions. His work remains a foundational part of modern topology and geometry, and the geometric perspective he championed continues to shape research in these fields.