Alexander Grothendieck, whose gift for deep abstraction excavated new ground in the field known as algebraic geometry and supplied a theoretical foundation for the solving of some of the most vexing conundrums of modern mathematics, died on Thursday in Ariège, in the French Pyrenees. He was 86.

A vexing character himself, Mr. Grothendieck (pronounced GROAT-en-deek) turned away from mathematics at the height of his powers in the early 1970s and had lived in seclusion since the early 1990s. His death was widely reported in France, where the newspaper Le Monde described him as “the greatest mathematician of the 20th century.” In a statement on Friday, President François Hollande praised him as “one of our greatest mathematicians” and “an out-of-the-ordinary personality in the philosophy of life.”

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.

“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.”