Vladimir Voevodsky, a mathematician who grew up in Russia before coming to the United States, where his work was recognized with the Fields Medal, often regarded as the equivalent of the Nobel Prize for mathematics, died Sept. 30 in Princeton, N.J. He was 51.
His death was confirmed by the Institute for Advanced Study, the research institution where he was a professor and had spent most of his working life. He reportedly collapsed in his home from a sudden aneurysm, and was discovered by friends.
Mr. Voevodsky received the Fields Medal, which is awarded for outstanding contributions by mathematicians under the age of 40, in 2002 in recognition of work in the area mathematicians call homotopy.
"His amazing insight," Chris Kapulkin, a mathematician at the University of Western Ontario, said in an interview, "was that at some very high level in both geometry and algebra you can see the same structure. You can transfer results from one to another."
As a pure mathematician, Mr. Voevodsky possessed powers of imagination, visualization and reasoning that could be applied at levels of abstraction almost impossibly remote from the minds and lives of most people.
Aware of this, he was also troubled by it. "I cannot explain — even to a very good student in his final year at university — the details of my work!" he once said in an interview. "Today, new people find it harder and harder to engage in the scientific process. I think it's a bad sign."
Uneasy about estranging himself too far from the real world of useful applications, he pursued work in his later years that was applicable to improving the process by which mathematical proofs were produced.
Some of his work was in the area of developing "proof assistants," which entailed harnessing the computer to help mathematicians through the dauntingly intricate steps by which abstruse proofs are developed.
In working to develop the proof assistant, Mr. Voevodsky appeared to acknowledge an episode from his career in which an important result was later challenged. In 2013, after years spent in checking, he concluded he had erred. Other mathematicians lauded his honesty and saw his admission as a sign they needed to avail themselves of the calculating capacity of computers.
"I didn't have the tools to explore the areas where curiosity was leading me and the areas that I considered to be of value and of interest and of beauty," Mr. Voevodsky once wrote. "So I started to look into what I could do to create such tools."
Soon, he said, it became clear the only path to follow was "somehow to make it possible for me to use computers to verify my abstract, logical, and mathematical constructions."
Some of the work, intended to enable computers to work in pure mathematics, entailed providing a new formulation of the foundations of the ancient discipline.
Arguments have sprung up as to what the proper role of computers in proofs should be. Mr. Voevodsky's position was they assisted in proofs, but did not produce them.
In fact, he said, they reminded him of what the true nature of proof, so fundamental to mathematics, ought to be.
Unique in their abilities to explore realms of the mind few of their fellows could enter, mathematicians often show unique traits in their personal lives. Mr. Voevodsky was no exception.
Vladimir Alexandrovich Voevodsky was born on June 4, 1966, in Moscow. Both of his parents were scientists. He enrolled at Moscow State University but he was bored by his classes and expelled for what he called "academic failure."
Ultimately, he went back to school, but as the Soviet Union began to deteriorate, "everything collapsed, and such formalities as degrees seemed quite useless."
He collaborated on academic papers with another mathematician, Mikhail Kapranov, who came to the United States for graduate school and mentioned his friend.
Soon, without a college degree and without formally applying, as he told it, Mr. Voevodsky was a graduate student at Harvard University. But he was robbed in Boston and had other difficulties adjusting, and he temporarily went back to Russia.
Harvard kept his fellowship open, and he lived in his office for a time before receiving his doctorate in 1992.
The Institute for Advanced Study in Princeton listed him as a member in the 1992 -1993 academic year. He was also a member, according to the institute,from 1998 to 2001. In January of 2002 he became a member of the mathematics faculty at the institute, one of the world's most renowned research institutions (one of the Institute's first faculty members was Albert Einstein, who was there from 1933 until he died in 1955).
Mr. Voevodsky also held appointments at other academic institutions. He was an associate professor at Northwestern University from 1997 to 1998, and a visiting scholar at the Max-Planck-Institute in Germany in 1996 and 1997. He was affiliated with Harvard from 1993 to 1997; first as a junior fellow in the Harvard Society of Fellows, then as a visiting scholar.
In 2002, the International Mathematical Union awarded Mr. Voevodsky the Fields Medal. According to the Institute for Advanced study, he was awarded the medal for development of new cohomology theories for algebraic varieties. He was recognized as the creator of motivic cohomology.
He received many fellowships and National Science Foundation grants for his work.
Among the fields in which he made contributions are the homotopy theory of schemes, algebraic K-theory, and interrelations between algebraic geometry and algebraic topology.
He was credited with developing solutions to what mathematicians know as the Milnor and Bloch-Kato conjectures.
His "Univalence Axiom" was also credited with making a strong impact in both mathematics and computer science.
His marriage to Nadia Shalaby ended in divorce. Survivors include two daughters.