Math is still quite hard: The headline shouted at us from yesterday’s Washington Post:
Reclusive genius a ‘giant’ of math
In an 1100-word piece, Matt Schudel reported the death, at age 86, of Alexander Grothendieck, “whose brilliant mind electrified the world of mathematics in the 1950s and 1960s, earning him the equivalent of the Nobel Prize in his field.”
We’d never heard of Grothendieck! Hungrily, we fell upon the Post report, hoping to learn what his brilliant mind had done.
Basically, Schudel wasn’t saying. He started by telling us this:
SCHUDEL (11/16/14): Mr. Grothendieck (pronounced GROHT-en-deek) emerged from a life of exile during World War II to become one of the most important mathematical thinkers of the 20th century. His contributions to mathematics were often likened to those of Albert Einstein in physics.We were still a bit unclear. In what way did Grothendieck transform mathematics entirely?
His nominal specialty was algebraic geometry, which combines elements of both mathematical disciplines, but Mr. Grothendieck used his remarkable capacity for abstract thinking to make advances across the entire spectrum of mathematics.
He developed unifying concepts that could be applied to a variety of avenues of mathematical thought, including number theory, category theory, functional analysis and topology.
In 1966, Mr. Grothendieck was awarded the Fields Medal, considered the world’s highest honor in mathematics. Two of his major publications, “Elements of Algebraic Geometry” and “Fundamentals of Algebraic Geometry,” are so essential to mathematicians that they are known simply by their initials in French, EGA and FGA.
“He was one of the giants of mathematics, who transformed mathematics entirely with his work,” Cedric Villani, who won the Fields Medal in 2010, told Agence France-Presse.
Schudel soldiered on:
SCHUDEL (continuing directly): As a student, Mr. Grothendieck once recalled, he was taught how to calculate the volume of a sphere and other geometric shapes, but he sought a deeper understanding: the definition of volume itself.He sought a deeper understanding—the definition of volume itself? We don’t understand that either.
When he embarked on his career, he didn’t concentrate on solving age-old puzzles so much as on developing new, simplified approaches to mathematical investigation. Other scholars came to apply Mr. Grothendieck’s theoretical frameworks to such fields as computer programming, software development, satellite communications, classification systems and the study of biological data.
His ideas were instrumental in solving one of the enduring conundrums of mathematics, Fermat’s Last Theorem. In 1637, Pierre de Fermat jotted a mathematical notation in the margin of a book, but its proof had baffled the world’s greatest mathematical minds for more than three centuries.
Finally, in 1995, British mathematician Andrew Wiles published a proof of the theorem. He arrived at his solution using the principles of algebraic geometry, the field that Mr. Grothendieck had redefined to its foundations.
As we said, Schudel devoted 1100 words to his obituary. When we were done, we had no real idea what Grothendieck had actually done, mathematically speaking.
Luckily, Sunday’s New York Times also featured a long obituary. Instantly, Weber and Rehmeyer tackled the problem:
WEBER AND REHMEYER (11/16/14): Algebraic geometry is a field of pure mathematics that studies the relationships between equations and geometric spaces. Mr. Grothendieck was able to answer concrete questions about these relationships by finding universal mathematical principles that could shed unexpected light on them. Applications of his work are evident in fields as diverse as genetics, cryptography and robotics.We don’t understand that either. Math remains very hard.
“He had an extremely powerful, almost otherworldly ability of abstraction that allowed him to see problems in a highly general context, and he used this ability with exquisite precision,” Allyn Jackson wrote in a 2004 biographical essay about Mr. Grothendieck for Notices of the AMS, a journal of the American Mathematical Society. “Indeed, the trend toward increased generality and abstraction, which can be seen across the whole field since the middle of the 20th century, is due in no small part to Grothendieck’s influence.”