Alexander Grothendieck
Years:
Title: Greatest Algebraic Geometer of the 20th Century
Alexander Grothendieck
Born: 1928
Died: 2014
Title: Greatest Algebraic Geometer of the 20th Century
The Hermit with "God's Perspective"
In the second half of the 20th century, Grothendieck completely reconstructed algebraic geometry.
He was so abstract that some say he viewed mathematics from "God's perspective." He reconstructed algebraic geometry with such radical abstraction that his work remains difficult even for experts. Yet his ideas transformed mathematics and physics.
Core Contributions - Deep Analysis
Scheme Theory: A New Foundation for Geometry
Grothendieck's schemes generalize algebraic varieties to work over any ring, not just fields. This abstraction was so radical that:
- It unified number theory, geometry, and algebra
- It made previously impossible proofs possible
- It connected mathematics to physics (string theory)
The Grothendieck-Riemann-Roch Theorem
A fundamental result connecting topology, geometry, and algebra. This theorem shows how to compute invariants of algebraic varieties—a problem that seemed intractable before Grothendieck.
Étale Cohomology
Grothendieck developed new cohomology theories that work in characteristic (where traditional methods fail). This was crucial for:
- Number theory: Proving the Weil conjectures
- Algebraic geometry: Understanding varieties over finite fields
- Physics: Applications in string theory
The Reclusive Genius
Grothendieck was born to anarchist parents, spent his childhood in concentration camps, and became one of the greatest mathematicians of the 20th century. But in 1991, he disappeared.
Anecdote: At the peak of his career, he suddenly withdrew from mathematics due to anti-war beliefs, secluding himself at the foot of the Pyrenees, cutting off all contact with the world until his death.
He left mathematics, burned thousands of pages of unpublished work, and lived as a recluse in the Pyrenees. He refused to communicate with the mathematical community, even declining the Fields Medal and other honors.
Why He Left
Grothendieck was deeply troubled by:
- Militarization of science: He saw mathematics being used for weapons
- Academic politics: He was disillusioned with the mathematical establishment
- Spiritual crisis: He sought meaning beyond mathematics
His disappearance was one of the great mysteries of 20th-century mathematics.
Legacy
Despite his reclusiveness, Grothendieck's work transformed mathematics:
- Algebraic Geometry: Completely rebuilt the field
- Number Theory: Proved the Weil conjectures (via Deligne)
- Topology: New cohomology theories
- Physics: Foundation for string theory and mirror symmetry
Grothendieck showed that radical abstraction can reveal deep truths. His work is so fundamental that modern algebraic geometry is essentially "Grothendieck's geometry." Yet he chose to walk away from it all, leaving behind a mathematical revolution that continues to shape mathematics today.