How Alexander Grothendieck Revolutionized 20th-Century Mathematics | Quanta Magazine
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How Alexander Grothendieck Revolutionized 20th-Century Mathematics
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How Alexander Grothendieck Revolutionized 20th-Century Mathematics
By
Konstantin Kakaes
May 20, 2026
Grothendieck is revered in the world of math; outside of it, he’s known for his unusual life, if he’s known at all. But what were his actual mathematical contributions?
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Mercedes deBellard for Quanta Magazine
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By Konstantin Kakaes
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May 20, 2026
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What Albert Einstein was to 20th-century physics, Alexander Grothendieck was to 20th-century mathematics. He is much less well known because math gets technical even more quickly than physics does. But as with Einstein, Grothendieck’s impact came not just from his own results, revolutionary though they were. His work also reoriented his entire discipline in radical new directions.
Grothendieck was intense and ascetic from his early days. Starting in the early 1950s, when he was in his 20s, he produced thousands of pages of formal and informal notes that changed the course of mathematics. Then in 1970, he quit. He left his post at a prestigious research institute just outside of Paris to teach at the provincial university in Montpellier where he studied as an undergraduate. He mostly stopped talking to other mathematicians. In the early 1990s, he moved to a small village in the Pyrenees, where he lived as a hermit.
Mathematicians are still grappling with the innovations he made half a century ago. His work pushed mathematics to a new level of abstraction by focusing on the relationships between objects rather than the objects themselves. “If there is one thing in mathematics which fascinates me more than any other (and undoubtedly always has), it is neither ‘number’ nor ‘size,’ but invariably shape,” he wrote in his memoirs. “And among the thousand and one faces under which shape chooses to reveal itself to us, that which has fascinated me more than any other and continues to do so is the structure hidden in mathematical things.”
His revolutionary mathematics centered around that search for hidden structure.
Revealing Shapes
Grothendieck is most famous for his work in algebraic geometry. The field first developed as the study of shapes defined by polynomial equations — equations that add together variables raised to fixed powers. These can be as simple as a line (x – y = 0) or a circle (x2 + y2 – 1 = 0). But as you consider more and more variables raised to higher powers and also look for solutions that satisfy sets of many equations instead of just one, things quickly get more complicated — and more abstract.
Grothendieck, seen here in 1954, was fascinated by hidden geometric structure. “If there is one thing in mathematics which fascinates me more than any other (and undoubtedly always has), it is neither ‘number’ nor ‘size,’ but invariably shape,” he wrote.
Paul R. Halmos photograph collection, e_ph_08592_pub, The Dolph Briscoe Center for American History, The University of Texas at Austin
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The discipline took flight in the late 19th century, when mathematicians started asking questions about what happens if instead of plugging ordinary numbers into your equations, you plug in numbers from other, more abstract sets.
Before Grothendieck, algebraic geometry was an interesting and vibrant subdiscipline within mathematics. But it was also somewhat in crisis, as the mathematician David Mumford later wrote. “Every researcher used his own definitions and terminology, in which the ‘foundations’ of the subject had been described in at least half a dozen different mathematical ‘languages.’”
Then “Grothendieck came along and turned a confused world of researchers upside down, overwhelming them with [a] new terminology … as well as with a huge production of new and very exciting results.”
Grothendieck came along and turned a confused world of researchers upside down.
David Mumford
Grothendieck is most famous for introducing mathematical constructions that helped him and others prove longstanding conjectures, and that eventually became central objects of...