Stephen Smale

Stephen Smale
Born	Stephen Smale 15 7, 1930
Birthplace	Flint, Michigan, U.S.
Nationality	American
Occupation	Mathematician
Employer	University of California, Berkeley (Professor Emeritus)
Known for	Topology, dynamical systems, mathematical economics, higher-dimensional Poincaré conjecture
Education	University of Michigan (BS, PhD)
Awards	Fields Medal (1966), National Medal of Science (1996), Wolf Prize in Mathematics (2007)
Website	[http://math.berkeley.edu/~smale/ Official site]

Stephen Smale (born July 15, 1930) is an American mathematician whose work has shaped several major branches of modern mathematics, including topology, dynamical systems, and mathematical economics. Born in Flint, Michigan, Smale rose to international prominence for his proof of the higher-dimensional generalization of the Poincaré conjecture, a result that earned him the Fields Medal in 1966—mathematics' most prestigious honor for work accomplished before the age of forty. Over a career spanning more than six decades, he has made foundational contributions to differential topology, the theory of computation, and numerical analysis. Smale spent more than three decades on the mathematics faculty of the University of California, Berkeley, where he remains Professor Emeritus. His influence extends well beyond pure mathematics; his formulation of problems in economics, his celebrated list of eighteen unsolved problems for the twenty-first century, and his later work on learning theory and protein folding have drawn researchers across disciplines. His career has also been marked by political activism, particularly his outspoken opposition to the Vietnam War during the 1960s, which brought him into public controversy even as his mathematical reputation was ascending.

Early Life

Stephen Smale was born on July 15, 1930, in Flint, Michigan, a city then known as a center of the American automobile industry. He grew up in a working-class environment in Michigan during the Great Depression and the years of World War II. Details of his childhood are relatively sparse in the public record, but by the time he reached college age, he had developed a keen interest in mathematics and the sciences that would lead him to pursue higher education at the University of Michigan.

Smale's early intellectual development took place in an era when American mathematics was undergoing rapid expansion, fueled in part by the influx of European mathematicians fleeing political turmoil and by the postwar investment in scientific research. Michigan, with its strong tradition in mathematics, provided a natural setting for a young student with an aptitude for abstract thinking. Smale has spoken in various forums about the gradual nature of his mathematical awakening, noting that his path to becoming a research mathematician was not always straightforward.

Education

Smale enrolled at the University of Michigan, where he earned his Bachelor of Science degree in mathematics. He continued at the same institution for his graduate studies, working under the supervision of Raoul Bott, a distinguished mathematician known for his contributions to differential geometry and topology. Smale completed his doctoral dissertation in 1957, titled "Regular Curves on Riemannian Manifolds," which addressed questions in differential topology concerning the classification and behavior of curves on curved spaces.

The choice of Bott as his advisor proved formative. Bott's geometric intuition and broad perspective on mathematics influenced Smale's approach to problems, encouraging a style that combined topological insight with analytic technique. The University of Michigan's mathematics department during the 1950s was a fertile environment, and Smale benefited from exposure to a community of mathematicians engaged with the latest developments in topology and related fields.

Career

Early Work in Topology and the Poincaré Conjecture

Following the completion of his doctorate, Smale began his research career with a series of results that quickly established him as one of the leading topologists of his generation. His earliest significant work concerned the problem of immersions of spheres—specifically, the question of how a sphere can be "turned inside out" in three-dimensional space without creating any creases or singularities, a result now known as Smale's sphere eversion theorem. This counterintuitive result, demonstrated in 1958, showed that a sphere could be smoothly everted, defying the expectations of many mathematicians at the time.

Smale's most celebrated achievement in topology came with his proof of the generalized Poincaré conjecture in dimensions five and higher, accomplished in 1960. The Poincaré conjecture, originally posed by Henri Poincaré in 1904, asked whether every simply connected, closed manifold is homeomorphic to a sphere. While Poincaré's original formulation concerned three-dimensional spaces, the question extends naturally to higher dimensions. Smale proved that the conjecture holds in all dimensions greater than or equal to five, using a technique he developed called the "h-cobordism theorem." This was a landmark result in differential topology; the h-cobordism theorem itself became a fundamental tool in the classification of manifolds and remains central to the field.

The proof was notable for its ingenuity and for the fact that higher-dimensional cases were resolved before the original three-dimensional conjecture, which would not be settled until Grigori Perelman's work in 2003. Smale's approach exploited the additional "room" available in higher dimensions to perform topological manipulations that are not possible in lower dimensions. The work was carried out in part while Smale was visiting the Instituto Nacional de Matemática Pura e Aplicada (IMPA) in Rio de Janeiro, Brazil—a circumstance that would later become a subject of public discussion.

Fields Medal and Political Controversy

In 1966, Smale was awarded the Fields Medal at the International Congress of Mathematicians in Moscow. The Fields Medal, often described as the highest honor in mathematics, is awarded every four years to mathematicians under the age of forty. Smale's medal recognized his proof of the generalized Poincaré conjecture and his broader contributions to differential topology.

The award ceremony in Moscow coincided with a period of intense political engagement for Smale. He was a vocal opponent of the Vietnam War and had been active in anti-war movements in the United States. His decision to accept the Fields Medal in Moscow, during the Cold War, attracted attention from both the mathematical community and the broader public. Reports have documented that the U.S. House Committee on Un-American Activities took an interest in Smale's political activities, and the National Science Foundation reportedly came under pressure regarding its funding of his research. Smale's willingness to combine his mathematical career with outspoken political views made him a distinctive figure in the academic world of the 1960s.

The story of Smale doing some of his most important mathematical work on the beaches of Rio de Janeiro became one of the most frequently retold anecdotes in the history of modern mathematics. The image of a mathematician proving a major theorem on a beach captured the public imagination and was referenced in discussions about the nature of mathematical creativity and the environments in which breakthroughs occur.

University of California, Berkeley

Smale joined the faculty of the University of California, Berkeley in 1960–1961 and returned permanently in 1964. He would remain at Berkeley for more than three decades, until 1995, making it the primary base for the most productive period of his career.

At Berkeley, Smale's research interests broadened considerably. While he continued to make contributions to topology, he turned increasingly to the study of dynamical systems—the mathematical theory of systems that evolve over time according to fixed rules. His work in this area was transformative. Smale introduced the concept of the "horseshoe map" (now known as the Smale horseshoe), a geometric construction that provided one of the first clear, rigorous examples of chaotic behavior in a deterministic system. The horseshoe demonstrated how stretching and folding of a region in phase space could produce infinitely complex dynamics, including sensitive dependence on initial conditions—a hallmark of what would later be popularly known as "chaos theory."

The Smale horseshoe became one of the central objects of study in the theory of dynamical systems and provided a template for understanding how chaotic behavior arises in a wide variety of physical, biological, and economic systems. As Quanta Magazine noted in a 2022 article, tools developed by Smale and his contemporaries remain essential for mathematicians attempting to make sense of chaotic dynamical systems.

Smale also made significant contributions to mathematical economics during his Berkeley years. He applied techniques from differential topology and dynamical systems to problems in general equilibrium theory, exploring the structure and stability of economic equilibria. His work in this area influenced a generation of mathematical economists and helped establish deeper connections between pure mathematics and economic theory.

Algorithms, Computation, and Smale's Problems

From the 1980s onward, Smale increasingly focused on questions at the interface of mathematics and computer science, particularly the theory of algorithms and computational complexity. He studied the efficiency and convergence properties of algorithms used in numerical analysis and optimization, including Newton's method and related iterative schemes. His work in this area was recognized with a writing award from the Mathematical Association of America for his expository article "On the Efficiency of Algorithms in Analysis."

In 1998, Smale published a list of eighteen unsolved mathematical problems for the twenty-first century, often referred to as "Smale's problems." This list, inspired in part by David Hilbert's famous list of problems from 1900, encompassed a broad range of topics including the Poincaré conjecture (in dimension three), the Riemann hypothesis, the P versus NP problem, and questions in dynamical systems, numerical analysis, and mathematical economics. Several of these problems remain open, and the list has served as a guiding framework for research in multiple areas of mathematics.

Later Career: City University of Hong Kong and Toyota Technological Institute

After his formal retirement from Berkeley in 1995, Smale continued active research at several institutions. He held positions at the City University of Hong Kong, where he worked on problems in learning theory, approximation, and the mathematical foundations of data analysis. He also held a position at the Toyota Technological Institute at Chicago, a research institute affiliated with the University of Chicago that focuses on computer science and information technology.

During this period, Smale's research interests evolved to include machine learning, the geometry of data, and the protein folding problem. In a 2013 lecture at the Institute of Science and Technology Austria (ISTA), Smale presented work on "The Protein Folding Problem via Relations with Patterns in Data and the Geometry of Kernels," connecting ideas from computational geometry and kernel methods to one of the central challenges in computational biology. This work reflected Smale's characteristic willingness to move into new areas and apply mathematical tools to problems in the natural sciences.

Smale also maintained an affiliation with Columbia University and the University of Chicago at various points during his later career.

Personal Life

Stephen Smale is known to have pursued interests outside of mathematics, most notably as a collector of minerals and gemstones. His mineral collection, particularly of fine crystals, has been described as one of the most significant private collections of its kind. A book titled The Smale Collection: Beauty in Natural Crystals documents this collection. The collection has been exhibited and has drawn attention in the world of mineral collecting and natural history.

Smale's political activism, particularly during the 1960s, was a notable aspect of his public life. His opposition to the Vietnam War and his participation in protest movements brought him into conflict with governmental authorities and distinguished him from many of his academic contemporaries.

As of 2025, Smale remains Professor Emeritus at the University of California, Berkeley, and continues to be an active presence in the mathematical community.

Recognition

Stephen Smale has received numerous honors and awards throughout his career, reflecting the breadth and impact of his mathematical contributions.

• Fields Medal (1966): Awarded for his proof of the generalized Poincaré conjecture in dimensions five and higher and his contributions to differential topology.
• National Medal of Science (1996): The highest scientific honor bestowed by the United States government, recognizing his cumulative contributions to mathematics.
• Wolf Prize in Mathematics (2007): Awarded by the Wolf Foundation in Israel for his contributions to mathematics.
• Chauvenet Prize: Awarded by the Mathematical Association of America for outstanding mathematical exposition.

Smale was elected to the National Academy of Sciences and has received honorary degrees and visiting professorships from institutions around the world. His name is attached to several mathematical concepts and structures, including the Smale horseshoe, Smale's sphere eversion, and the Smale conjecture.

In recognition of his influence on the field, the Smale Prize was established to honor outstanding young researchers in the fields of the foundations of computational mathematics. Among the recipients of this prize is Shayan Oveis Gharan, a professor at the University of Washington, whose work in the theory of computation reflects the ongoing influence of Smale's research agenda.

In August 2025, a conference was organized in honor of Smale's 95th birthday, drawing mathematicians from around the world. Matthew Kvalheim, a mathematician at the University of Maryland, Baltimore County (UMBC), was among those invited, with his research described as "heavily influenced by Stephen Smale's breakthrough solution to a longstanding mathematical puzzle."

Legacy

Stephen Smale's influence on mathematics is reflected in the range of fields he has shaped and the tools and concepts that bear his name. His proof of the generalized Poincaré conjecture established new methods in differential topology that became standard in the field. The h-cobordism theorem, which emerged from this work, remains a cornerstone of the classification of manifolds and has been applied and generalized by subsequent generations of topologists.

In dynamical systems, the Smale horseshoe provided one of the first rigorous models of deterministic chaos and helped lay the groundwork for the modern mathematical study of chaotic systems. As Quanta Magazine observed, the tools that Smale and his contemporaries developed continue to be central to efforts to understand chaotic behavior across mathematics and the sciences. The horseshoe construction has become a standard example in textbooks and courses on dynamical systems and is a touchstone for researchers studying the structure of complex dynamical phenomena.

Smale's list of eighteen problems for the twenty-first century has served as a roadmap for mathematical research, analogous in ambition—if not in historical stature—to Hilbert's problems of 1900. Several of the problems on Smale's list have been resolved, including the three-dimensional Poincaré conjecture (proved by Perelman in 2003), while others remain open and continue to drive research.

His later work on algorithms, learning theory, and the mathematical foundations of data analysis anticipated many of the concerns that have come to dominate applied mathematics and computer science in the twenty-first century. By bringing the perspective and techniques of pure mathematics to bear on problems in computation, economics, and biology, Smale demonstrated the power of mathematical abstraction to illuminate questions across the sciences.

The collected papers of Stephen Smale, reviewed in the Notices of the American Mathematical Society, attest to the volume and diversity of his output, spanning pure topology, dynamical systems, economics, computation, and learning theory. His career stands as an example of the capacity of a single mathematician to contribute foundational results across multiple disciplines over the course of many decades.