He was born in
Moscow, the son of Alexander Lefschetz and his wife Sarah or Vera Lifschitz, Jewish traders who used to travel around Europe and the Middle East (they held
Ottoman passports). He moved towards mathematics, receiving a
Ph.D. in algebraic geometry from
Clark University in Worcester, Massachusetts in 1911. He then took positions at the
University of Nebraska and
University of Kansas, moving to
Princeton University in 1924, where he was soon given a permanent position. He remained there until 1953. In the application of topology to algebraic geometry, he followed the work of
Charles Émile Picard, whom he had heard lecture in Paris at the
École Centrale Paris. He proved theorems on the topology of
hyperplane sections of
algebraic varieties, which provide a basic inductive tool (these are now seen as allied to
Morse theory, though a
Lefschetz pencil of hyperplane sections is a more subtle system than a Morse function because hyperplanes intersect each other). The
Picard–Lefschetz formula in the theory of
vanishing cycles is a basic tool relating the
degeneration of families of varieties with 'loss' of topology, to
monodromy. He was an Invited Speaker of the
ICM in 1920 in Strasbourg. His book ''L'analysis situs et la géométrie algébrique'' from 1924, though opaque foundationally given the current technical state of
homology theory, was in the long term very influential (one could say that it was one of the sources for the eventual proof of the
Weil conjectures, through
SGA 7 also for the study of
Picard groups of
Zariski surface). In 1924 he was awarded the
Bôcher Memorial Prize for his work in
mathematical analysis. He was elected to the United States
National Academy of Sciences in 1925 and the
American Philosophical Society in 1929. The
Lefschetz fixed-point theorem, now a basic result of topology, was developed by him in papers from 1923 to 1927, initially for
manifolds. Later, with the rise of
cohomology theory in the 1930s, he contributed to the
intersection number approach (that is, in cohomological terms, the ring structure) via the
cup product and duality on manifolds. His work on topology was summed up in his monograph
Algebraic Topology (1942). From 1944 he worked on
differential equations. He was editor of the
Annals of Mathematics from 1928 to 1958. During this time, the
Annals became an increasingly well-known and respected journal, and Lefschetz played an important role in this. In 1945 he travelled to Mexico for the first time, where he joined the Institute of Mathematics at the
National University of Mexico as a visiting professor. He visited frequently for long periods, and during 1953–1966 he spent most of his winters in Mexico City. The RIAS mathematics group stimulated the growth of nonlinear differential equations through conferences and publications. He left RIAS in 1964 to form the Lefschetz Center for Dynamical Systems at
Brown University, Providence, Rhode Island. ==Selected works==