Solomon Lefschetz

Solomon Lefschetz
Name	Solomon Lefschetz
Caption	Lefschetz in 1968
Birth date	3 September 1884
Birth place	Moscow, Russian Empire
Death date	5 October 1972
Death place	Princeton, New Jersey, United States
Fields	Mathematics
Workplaces	University of Kansas, Princeton University
Alma mater	École Centrale Paris
Doctoral advisor	William Edward Story
Doctoral students	John Milnor, Clifford Truesdell, Richard Bellman
Known for	Lefschetz fixed-point theorem, Lefschetz hyperplane theorem, Lefschetz duality, Picard–Lefschetz theory
Awards	Bôcher Memorial Prize (1924), National Medal of Science (1964), Leroy P. Steele Prize (1970)

Solomon Lefschetz was a pioneering American mathematician whose profound work fundamentally shaped the landscape of algebraic topology and algebraic geometry in the 20th century. Born in the Russian Empire, he overcame a severe industrial accident to forge an illustrious career, primarily at Princeton University. His innovative use of topological methods to study algebraic varieties bridged major mathematical disciplines, earning him widespread recognition including the National Medal of Science.

Biography

Born in Moscow to Turkish Jewish parents, he moved to Paris for his education, graduating from the École Centrale Paris as an engineer. After emigrating to the United States, a tragic accident at the Westinghouse Electric Corporation led to the amputation of both his hands, after which he turned entirely to mathematics. He earned his PhD from Clark University under William Edward Story and later held positions at the University of Nebraska–Lincoln and the University of Kansas before joining the faculty at Princeton University in 1924. At Princeton, he became a central figure in the mathematics department, mentoring a generation of leading mathematicians and later serving as editor of the influential journal Annals of Mathematics.

Mathematical work

His research centered on applying the powerful tools of topology to classical problems in algebraic geometry. He developed what became known as the Lefschetz pencil technique, a method for studying the homology of complex algebraic varieties. His seminal text, *L'analysis situs et la géométrie algébrique*, laid the groundwork for modern intersection theory. This work deeply influenced subsequent developments by mathematicians like Hassler Whitney, René Thom, and Michael Atiyah, effectively creating the field of global analysis.

Lefschetz fixed-point theorem

A cornerstone of his legacy, the Lefschetz fixed-point theorem provides a powerful topological criterion for the existence of fixed points of a continuous mapping on a compact space. It generalizes earlier results like the Brouwer fixed-point theorem by relating the number of fixed points to the alternating sum of traces induced on homology groups. This theorem found immediate applications in differential equations and dynamical systems, and its generalizations became crucial in number theory, particularly within the Weil conjectures.

Lefschetz hyperplane theorem

Another fundamental result, the Lefschetz hyperplane theorem, describes the homology of a projective algebraic variety in relation to its hyperplane section. It states that for a smooth variety, the homotopy groups and homology groups up to the middle dimension are isomorphic to those of a hyperplane section. This theorem became an indispensable tool in complex geometry and inspired further deep results in Hodge theory, influencing the work of Kunihiko Kodaira and David Mumford.

Awards and honors

His contributions were recognized with numerous prestigious awards. He received the Bôcher Memorial Prize from the American Mathematical Society in 1924 for his work on analysis situs. Four decades later, he was awarded the National Medal of Science by President Lyndon B. Johnson. In 1970, he received the Leroy P. Steele Prize for his lifetime of seminal work. He was also elected to the National Academy of Sciences and served as president of the American Mathematical Society.

Legacy

His influence extends far beyond his published theorems. He revitalized algebraic geometry in the United States and trained a remarkable group of doctoral students, including John Milnor, Clifford Truesdell, and Richard Bellman. The Lefschetz duality theorem and Picard–Lefschetz theory remain central concepts. Institutions like the Solomon Lefschetz Center for Dynamical Systems at Brown University honor his name, and the Mathematics Research Center at the University of Wisconsin–Madison was once directed by him, cementing his role as a pillar of 20th-century mathematics.

Category:American mathematicians Category:Topologists Category:National Medal of Science laureates