Maxim Kontsevich: The Mathematician Who Made Physics Rigorous
In 1990 a 25-year-old Soviet researcher with no university degree sat in a lecture hall in Bonn and listened to Michael Atiyah — "an eminent British mathematician who spoke of wonderful things," as he later put it — describe an outrageous conjecture of Edward Witten's about the geometry of curves. Within days, on a boat excursion down the Rhine, Kontsevich was explaining to colleagues how he intended to prove it. He was invited back to Bonn for a full year. Two years later the proof was his doctoral thesis.
Khimki, Korean, and an Olympiad
He was born on 25 August 1964 in Khimki, just outside Moscow, into a family of exacting scholarship. His father, Lev Rafailovich Kontsevich, was a leading Korean specialist at the Russian Academy of Sciences' Institute of Oriental Studies and the author of the Kontsevich system, still the standard method for transliterating Korean into Cyrillic. His mother trained as an engineer; his elder brother Leonid went into computer imaging research in San Francisco. Kontsevich credits the brother, and "some very good books," with turning him toward mathematics and physics.
The books did their work early. At sixteen he placed second in the national Mathematical Olympiad, which won him admission to Moscow State University without entrance examinations. He studied there from 1980 to 1985 under a formidable faculty, above all Israil Gelfand. He was publishing at nineteen: "The growth of the Lie algebra generated by two generic vector fields" with A. A. Kirillov in 1983, followed by "Algebras of intermediate growth" with Kirillov and A. I. Molev.
The Student Who Left
In 1985 he walked away from Moscow State University without taking a degree and joined the Institute for Problems of Information Transmission at the Academy of Sciences, publishing on the Virasoro algebra and related structures through 1988. It is worth pausing on this: the man who would win the Fields Medal spent his early twenties as an undegreed researcher in a Soviet information-theory institute. He first visited the Institut des Hautes Études Scientifiques in 1988 and never forgot it.
Four Problems
The Bonn thesis, submitted in 1992 under Don Zagier's supervision, was titled "Intersection Theory on the Moduli Space of Curves and the Matrix Airy Function." It proved Witten's conjecture: that intersection numbers of stable classes on compactified moduli spaces of algebraic curves obey the Korteweg–de Vries integrable hierarchy — a statement linking the geometry of Riemann surfaces to the mathematics of shallow-water waves, which had no business being true. The reviewer Clifford Henry Taubes noted that "many of the steps in this proof exhibit Kontsevich's unique talent for combinatorial calculations."
He then did it three more times in five years. His 1993 paper on Vassiliev's knot invariants introduced what is now the Kontsevich integral, a universal invariant of knots and links; the reviewer Joan Birman wrote that he had "taken a very fresh and original look" at the subject and made a complicated theory "seem both natural and clear." He proved that every Poisson manifold admits a formal deformation quantization, supplying explicit formulas in the flat case and settling a foundational question in mathematical physics. With Yuri Manin he wrote "Gromov–Witten classes, quantum cohomology, and enumerative geometry" in 1994, and introduced the moduli space of stable maps that made rigorous counting of curves possible — the machinery behind theorems on rational curves in Calabi–Yau threefolds that proved decisive for mirror symmetry. He also wrote "Formal (non)commutative symplectic geometry," and later, with Alexei Belov-Kanel, proved the equivalence of the Dixmier and Jacobian conjectures.
Mirror Symmetry as Category Theory
At the 1994 International Congress of Mathematicians in Zürich he proposed the idea that now bears his stamp more than any other: homological mirror symmetry. String theorists had noticed that Calabi–Yau manifolds come in mirror pairs whose complex and symplectic structures appear to swap. Kontsevich's conjecture is that this is an equivalence of categories — that the derived category of coherent sheaves on X, an object of pure algebraic geometry, is equivalent to the derived Fukaya category of its mirror Y, an object of pure symplectic geometry. It converted a physicist's observed coincidence into a precise structural claim, and it has organised large parts of geometry ever since.
Bures-sur-Yvette
He passed through Harvard, the Institute for Advanced Study at Princeton, Rutgers, and a professorship at Berkeley in the mid-1990s, then declined to settle in America despite his brother being there. He took a permanent professorship at the IHES in Bures-sur-Yvette, the institution he had admired since 1988, and has stayed. He also holds a distinguished professorship at the University of Miami, took French citizenship in 1999, and holds both Russian and French passports.
The prizes accumulated: the Henri Poincaré Prize in 1997; the Fields Medal at the 1998 Berlin Congress, cited for work in algebraic geometry and algebraic topology; the Daniel Iagolnitzer Prize; the Crafoord Prize in 2008, shared with Edward Witten, "for their important contributions to mathematics inspired by modern theoretical physics"; the Shaw Prize in 2012; the Breakthrough Prize in Fundamental Physics the same year; the Breakthrough Prize in Mathematics in 2015; membership of the Paris Academy of Sciences and foreign associateship of the U.S. National Academy of Sciences.
Why Maxim Is Called a Genius
Two distinct capacities are at work, and they are rarely found together. The first is brute calculational power of a very particular kind — Taubes's phrase, "unique talent for combinatorial calculations," is a technical assessment, not a compliment, and it describes the machinery that got the Witten conjecture proved when nobody else could. The second is a structural instinct for what physicists are really saying. String theorists produced dazzling, non-rigorous claims throughout the 1980s and 1990s; Kontsevich's habit was to find the mathematical object those claims were secretly about, then state it in a form mathematicians could work with. Homological mirror symmetry is the purest example: a physical coincidence turned into a categorical equivalence. Birman's observation that he makes complicated theory "seem both natural and clear" points at the same faculty from another angle.
The honest counter-case has three parts. Much of the raw material was not his — Witten conjectured, Vassiliev defined the invariants, the physicists found mirror symmetry; Kontsevich's role was to make things rigorous, which is indispensable but derivative by construction. Second, his most celebrated single idea remains a conjecture: homological mirror symmetry is unproven in general more than thirty years on, and conjectures are cheap next to theorems. Third, "genius" is currency that circulates freely in mathematics, and the specific public testimonials to Kontsevich tend to praise technique and originality rather than to reach for the word. The defensible claim is narrower and still extraordinary: for roughly a decade he was the most reliable translator between theoretical physics and rigorous mathematics that either field had.
Legacy
Entire research programmes now run on his definitions — stable maps, the Kontsevich integral, deformation quantisation, the Fukaya-category framing of mirror symmetry. He remains at the IHES, still working, and the conjecture he floated on a Zürich stage in 1994 continues to generate more mathematics than most completed proofs.
Achievements
- Fields medal — 1998
- Knight of the Legion of Honour
- Henri Poincaré Prize — 1997
- Breakthrough Prize in Mathematics — 2014
- Breakthrough Prize in Fundamental Physics — 2012
- Notable work: Kontsevich invariant
- Held posts at Rutgers University, Institut des Hautes Études Scientifiques and University of California, Berkeley
- Fields: mathematics
