Solomon Lefschetz

Solomon Lefschetz (1884-1972) did pioneering work in algebraic geometry, algebraic topology, and differential equations, and exerted tremendous influence over American mathematics as a professor at Princeton University and the editor of the journal Annals of Mathematics.

Lefschetz was born in Moscow into a Jewish family. His father had frequent business in Persia, and usually based his family in Paris. At age 21 Lefschetz emigrated to the United States. He worked briefly at the Baldwin Locomotive Works, then for Westinghouse Electric Company in Pittsburg from 1907 to 1910. His industrial career was cut short by an accident in which he lost both his hands. Lefschetz started over as a mathematician, receiving his doctorate from Clark University in 1911.

Lefschetz accepted a position at the University of Nebraska, then moved to the University of Kansas. Despite heavy teaching demands and near-complete isolation from other research mathematicians, he produced research papers of striking originality and importance. (He later wrote that his position "enabled me to develop my ideas in perfect mathematical calm".) In 1924 he went to Princeton as a visiting professor, and his post was made permanent the next year. Lefschetz became Henry Fine research professor in 1932 and retained that post until his retirement from Princeton in 1953.

Lefschetz began his research by studying algebraic varieties (sets defined by the vanishing of polynomials). He applied to them the ideas of algebraic topology invented by Poincaré, and developed algebraic intersection theory. Later he recast this intersection theory as the "cup product" in the cohomology theory developed by Alexander, Cech and Whitney, and this is the form in which it appears in his 1942 monograph Algebraic Topology.

One of Lefschetz's most widely used results, the Lefschetz fixed-point theorem, asserts that a map f from a "nice" compact space to itself has a fixed point (a point x such that f(x)=x) when a certain numerical invariant (the "Lefschetz number") of f is nonzero. Since the Lefschetz number depends only on the function f induces on the homology groups of the space, it is effectively computable. For some spaces (for example, those having the same homology groups as a point) the Lefschetz number of any map f from the space to itself is nonzero, so the Lefschetz fixed-point theorem shows that these spaces have the fixed-point property (any map from the space to itself has a fixed point).

After 1942 Lefschetz shifted his attention to differential equations. He actively pursued the geometric (or "qualitative") approach to nonlinear differential equations (again following in the footsteps of Poincaré), and established new results on the stability of equilibrium points and periodic orbits. After retiring from Princeton he became a consultant to the Research Institute for Advanced Studies (R.I.A.S.), an industry-sponsored research center. In 1964 the part of R.I.A.S. devoted to differential equations found a home at Brown University, where it became the Lefschetz Center for Dynamical Systems, with Lefschetz as visiting Professor of Applied Mathematics.

Lefschetz was famous for his intuitive style of reasoning and strong opinions. Students said of him that he never gave an incorrect result or a correct proof. His judgements could be harsh: for example, he despised most point-set topology as "baby stuff". He could be wrong, most famously in the case of a paper of William Hodge. Lefschetz thought that Hodge's paper was wrong, and told all his mathematical friends so. Hodge came to Princeton and gave a seminar on the paper (as Hodge later wrote, "owing to [Lefschetz's] characteristic interruptions the seminar actually lasted for six sessions"); afterward, Lefschetz publically acknowledged that Hodge was right and wrote to his friends admitting as much.

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