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, and who later disappeared into a mysterious life of self-imposed isolation, died Nov. 13 at a hospital in Saint-Girons, France. He was 86.

French media reports announced his death. The cause was not disclosed.

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.

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.

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.

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.

The circumstances of Mr. Grothendieck’s birth and childhood are in some doubt, shrouded by the dramas unfolding throughout Europe in the early 20th century.

His father was a Russian-born Jewish anarchist named Alexander Schapiro or possibly Morris Shapiro, who may have been imprisoned for attempting to assassinate the czar.

According to different accounts, he was either freed from prison by the Bolsheviks or was arrested by Bolsheviks during the Russian Revolution. Somehow, he managed to escape from Russia despite losing an arm.

He turned up in Berlin in the 1920s as a photographer using the name Alexander Tanaroff. He fell in love with a married woman named Johanna “Hanka” Grothendieck and told her husband, “I will steal your wife.”

Their illegitimate son was born in Berlin on March 28, 1928. His name at birth was Alexander Raddatz, the name of Johanna’s husband at the time. The child’s parents later married, but as the Nazi regime took over Germany in 1933, they fled to France.

Young Alexander, however, stayed behind with friends in Hamburg, Germany, until 1939, when he was reunited with his parents in Paris. His father was arrested and sent to the Auschwitz concentration camp, where he died in 1942.

Mr. Grothendieck took his mother’s maiden name and lived in various relocation camps until the end of World War II. He attended college in Montpellier, France, while making a name for himself in mathematics. He studied for a doctorate at a university in Nancy, France, but it is unclear whether he received the degree.

He taught in Brazil and at the University of Kansas in the 1950s before taking a position at the French Institute for Advanced Scientific Study near Paris in the late 1950s.

Colleagues noted that he owned very few books, which forced him to solve mathematical problems by original and unorthodox methods.

Mr. Grothendieck was a charismatic teacher and was guided by a strong sense of social conscience throughout his life. He was fascinated by physics, but he chose to study mathematics when he saw the effects of nuclear weapons, which were made possible by advanced physics.

When he received the Fields Medal in 1966, he refused to go to the awards ceremony in Moscow as an act of protest against Soviet militarism and imprisonment of dissident writers. He also opposed U.S. involvement in the Vietnam War and, in the late 1960s, traveled to Hanoi to lead lectures on mathematics.

In 1970, he resigned from his academic post when he learned that the institute received funding from the French defense ministry. He later taught at the University of Montpellier, but Mr. Grothendieck grew increasingly reclusive and eccentric.

He rejected the Swedish Crafoord Prize in mathematics in 1988, then abruptly broke off communication with colleagues and family in 1991.

He lived in small French villages, often sleeping on the floor and susbsisting on a strict vegetarian diet. He obsessively wrote thousands of pages of reflections on the nature of evil and transcribed his dreams as an effort to prove the existence of a divine being.

Scattered among the writings were occasional insights into new realms of mathematical inquiry, some of which were published and inspired a new generation of mathematicians.

Mr. Grothendieck was believed to have been married at least twice and to have had at least four children.

From time to time, he emerged from his private exile to correspond with colleagues. In a letter to Welsh mathematician Ronnie Brown, Mr. Grothendieck wrote about why mathematics was important: It allows people to do difficult things — and it creates the tools by which difficult things can be made simple.