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 is most famous for introducing mathematical constructions that helped him and others prove longstanding conjectures, and that eventually became central objects of study in their own right.

His work also put algebraic geometry in the center of a web of many other areas of math — among them topology, number theory, representation theory, and logic. “Grothendieck never worked directly in number theory,” said Brian Conrad of Stanford University, “but the ideas he introduced into algebraic geometry totally transformed how number theory is done.”

His first major result in algebraic geometry was his 1957 generalization of the Riemann-Roch theorem, a proof from a century earlier that dictates how the shape of a surface limits which functions can be defined on it. As Leila Schneps of the French National Center for Scientific Research wrote, Grothendieck’s proof “propelled him to instant stardom in the world of mathematics.”

Thanks to his techniques, “a whole new wealth of operations becomes available,” Conrad said. “It opens up a whole new way to think about why the theorem is true.”

Then, just as quickly, Grothendieck moved on to the next thing. At the 1958 International Congress of Mathematicians, he announced his intention to remake all of algebraic geometry. He was going to do it with something called a scheme.

A decade earlier, the mathematician André Weil had conjectured a link between solutions to polynomial equations defined in two very different mathematical settings. The first was finite fields, number systems that operate according to a cyclical form of arithmetic. The second was complex numbers, which take our familiar, everyday numbers and add the square root of -1, called i.

Weil made four conjectures that related polynomials from one setting to those from the other. These conjectures, Conrad said, “sound like communication between parallel universes.”