Alexander Grothendieck

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Alexander Grothendieck (1928–2014) was a German-born French mathematician who reshaped modern algebraic geometry and influenced many areas of mathematics.

Early life
- Born in Berlin on March 28, 1928, Grothendieck grew up as a stateless refugee. His father was a politically active anarchist and his mother was a journalist from Hamburg.
- During World War II he and his family faced internment and hiding; he survived by being sheltered in Le Chambon-sur-Lignon and other places.
- After the war, he studied mathematics in France, first in Montpellier and then in Nancy, where he earned his doctorate in 1953 under Laurent Schwartz and Jean Dieudonné.
- His early work was in functional analysis, where he became a leading expert on topological vector spaces.

Rise to influence
- In 1958 he joined the newly established Institut des hautes études scientifiques (IHÉS) and soon became its central figure.
- Grothendieck revolutionized algebraic geometry by introducing highly abstract, unifying ideas. He created the theory of schemes, which made the field much more flexible and powerful.
- He developed topos theory (a generalization of topology) and advanced cohomology theories, including étale cohomology, which were crucial for solving deep problems in number theory and geometry.
- He introduced abelian categories and showed that many geometric and algebraic questions could be understood as problems in homological algebra. This led to the reformulation of sheaf cohomology as derived functors.
- His work also produced foundational tools like the Grothendieck–Riemann–Roch theorem, K-theory, and the broad program behind Éléments de géométrie algébrique (EGA) and Séminaire de géométrie algébrique (SGA).
- The ideas Grothendieck developed at IHÉS helped prove major results such as the Weil conjectures, with his student Pierre Deligne completing the last steps. His influence extended far beyond algebraic geometry into number theory, topology, and logic.

Personal and political life
- Grothendieck was known for his radical, pacifist views. He opposed military funding and, in 1970, left IHÉS over funding issues related to the military.
- He turned increasingly to philosophy, spirituality, and social activism, founding a political group focused on antiwar and ecological concerns.
- He never stopped thinking about mathematics, but he moved away from publishing in traditional journals. He produced important manuscripts in seclusion, including Esquisse d'un Programme (Sketch of a Programme) and Pursuing Stacks, which helped shape later work in homotopy theory and category theory.
- His major autobiographical work Récoltes et Semailles (Harvests and Sowings) reflects on his life in mathematics and his relationship with the mathematical community. He declined the Crafoord Prize in 1988, arguing that money should not drive science.

Later years and legacy
- In his later life, Grothendieck lived quietly in the Pyrenees and later in a small village in France, continuing to think deeply about mathematics and philosophy.
- He left a vast amount of unpublished material, much of which is housed at the University of Montpellier and has been made accessible in various ways.
- Today, Grothendieck is widely regarded as one of the greatest mathematicians of the 20th century. His work on schemes, topos theory, étale cohomology, and categorical methods transformed how mathematicians approach geometry, number theory, and logic. His ideas created a lasting framework that continues to influence many areas of mathematics.

Personal life
- Grothendieck had five children and was involved with several partners over the years.
- He became French citizen later in life, but his early years were marked by statelessness and refugee status.

Alexander Grothendieck passed away on November 13, 2014, leaving a profound and lasting impact on mathematics.