Vladimir Voevodsky (1966–2017) Mathematician Who Revolutionized Algebraic Geometry and Computer Proof

OBITUARY COMMENT Vladimir Voevodsky (1966–2017) Mathematician who revolutionized algebraic geometry and computer proof.

ladimir Voevodsky revolutionized Voevodsky dreamed of a global repository algebraic geometry and is best of mathematical statements and proofs. This known for developing the new field would help mathematicians to accomplish, Vof ‘motivic homotopy theory’. His contribu- verify and share their work. tions to computer formalization of proofs By 2006, he had selected ‘type theory’ as and the foundations of mathematics also the appropriate formal language for such made an immense impact. a repository — it classifies mathemati- Algebraic geometry is the study of geomet- cal objects into ‘types’, such as triangles or ric aspects of systems of polynomial equa- curves. Voevodsky considered it a more tions, such as the equation x2 + y2 = 1, which natural language than the set theory con- yields a circle when x and y are real num- ventionally used by mathematicians. The bers, and something sharing the topologi- system of concepts he imagined would cal flavour of a circle when x and y are more organize types into an infinite hierarchy, abstract sorts of numbers. Voevodsky joins a with propositions at level 1, sets of things line of great mathematicians, including Bern- (such as natural numbers) at level 2, col- hard Riemann and Alexander Grothendieck, lections of structures (such as triangles) at who built algebraic geometry into a deep and level 3, and so on. A mechanism called uni- powerful science over the past two centuries. valence would allow mathematicians to use He died in September, aged 51, at his home in each other’s work even if they had different Princeton, New Jersey. approaches to the same underlying concepts. Voevodsky was born in Moscow in 1966; By 2010, he was ready to try out his ideas his father was a physicist and his mother was on the computer. Within three months, he a chemist. At first, he studied chemistry; but succeeded in developing a library of thou-

ANDREA KANE/INSTITUTE FOR ADVANCED STUDY ADVANCED FOR ANDREA KANE/INSTITUTE to understand it, physics was required, so he sands of pieces of code for his basic defini- began to study physics; but for it, mathemat- tions and theorems. He called this repository ics was required, so he began to study math- mathematics were music, then Voevodsky Foundations. ematics. He then attended Moscow State would be a musician who invented his own The system, which he named univalent University, but later stopped going to lectures key to play in. foundations, was the main topic of study at and thus received no degree. Nonetheless, the In 1996, less than four years after earning the Institute for Advanced Study for a year, in mathematics papers he had published were his doctoral degree, Voevodsky announced 2012–13. The Foundations library was incor- so promising that he was accepted to Harvard a proof of a famous 1970 conjecture formu- porated into a larger one called UniMath, University in Cambridge, Massachusetts, as lated by John Milnor — a stunning first con- whose aim is to formalize a substantial body a graduate student without applying (and firmation of the power of his ideas. For this, of mathematics. At his death, Voevodsky left without having even heard of it), earning his Voevodsky was awarded the Fields Medal, eight papers in various stages of completion PhD in 1992. the premier award in mathematics, in 2002, aimed at justifying the soundness of his new The next ten years were a period of high and was appointed professor in the School system. Univalent Foundations provides the productivity for him, during which he also of Mathematics at the Institute for Advanced basis for a global mathematics repository married and started a family. He made major Study in Princeton. Further work by him and and offers the first potentially viable alterna- progress towards Grothendieck’s grand others provided proofs for three other impor- tive to set theory as a foundation for all of vision, articulated in the 1960s, of a theory of tant conjectures. mathematics. ‘motives’. Grothendieck’s dream was to pro- Voevodsky was a visionary and meticu- Motivic homotopy theory is blossoming, duce, for any system of polynomial equations, lous mathematician, driven by an indomi- despite Voevodsky’s change of focus about the essential nugget that would remain after table will, but always gentle, friendly and ten years ago. Many dedicated research- everything apart from the shared topological open with those who met him. He achieved ers continue to find new ways to apply his flavour of the system was washed away. Per- much, despite being plagued by depression fundamental ideas to algebra, geometry haps borrowing the French musical term for for most of his life. He envisioned projects on and topology. Similarly, Univalent Founda- a recurring theme, Grothendieck dubbed this a grand scale, with multiple components that tions is destined to remain a vibrant area of the motif of the system. could be tackled one at a time over several research. Formalizing Voevodsky’s work on In Voevodsky’s motivic homotopy theory, years. Yet he was extremely adaptable. After motives in the Univalent Foundations would familiar classical geometry was replaced by success in one field, or finding that a line close the circle in a fitting way and fulfil one homotopy theory — a branch of topology of research was flawed, he soon moved on. of his dreams. ■ in which a line may shrink all the way down In 1997, for example, he turned to artificial to a point. He abandoned the idea that maps intelligence in robot locomotion, and later Daniel R. Grayson is professor emeritus of between geometric objects could be defined to mathematical biology. mathematics at the University of Illinois. He locally and then glued together, a concept In 2002, Voevodsky began thinking was a friend of Voevodsky from 1994 and that Grothendieck considered to be fun- about the computer representation of math- worked with him on computer proof checking. damental. A colleague commented that if ematical proofs. Like others before him, e-mail: [email protected]