Maxim Kontsevich

How to use this course

Maxim Kontsevich takes ideas from theoretical physics, which are often heuristic, and turns them into mathematics with exact statements. Many of his most influential contributions are conjectures and frameworks rather than single theorems: homological mirror symmetry, the formality of Hochschild cochains, motivic integration, the wall-crossing formula. Each has generated a research field. This course presents seven of them, alongside the story of an unusual career.

Each technical lesson opens with a paragraph for the general reader, then gives precise statements, proof sketches and exercises with folded solutions. Several lessons connect to the Witten module, because Kontsevich’s first great theorem proved a conjecture of Edward Witten, and the two shared the Crafoord Prize in 2008.

Timeline

Moscow, 1964–1985

Maxim Lvovich Kontsevich was born on 25 August 1964 in Khimki, a town just north-west of Moscow. He has said that he became seriously interested in mathematics at about ten or eleven. In his last years at secondary school in Moscow he took the advanced mathematics and physics courses that the Soviet system offered talented pupils, and at sixteen he placed second in the All-Union Mathematical Olympiad.

His olympiad result gave him entry to the Faculty of Mechanics and Mathematics of Moscow State University without the usual examinations. Moscow mathematics was then at a high point. Israel Gelfand’s seminar, among others, brought together analysis, representation theory and mathematical physics, and it trained students to treat these as one subject. Kontsevich attended Gelfand’s lectures.

He began research early. In 1983, while still a student, he published a paper with Alexander Kirillov on the growth of Lie algebras generated by vector fields. Infinite-dimensional Lie algebras built from geometry recur throughout his later work.

The research institute, 1985–1990

In 1985 Kontsevich left Moscow State University and took a research position at the Institute for Information Transmission Problems of the Soviet Academy of Sciences. Despite its name, the institute was home to a number of strong mathematicians working outside the university system.

In the second half of the 1980s, Moscow mathematicians and physicists were working intensively on conformal field theory, string theory and integrable systems, and seminars brought the two communities together. Kontsevich’s later work shows what he took from this setting: he reads Feynman diagrams as combinatorial objects with geometric meaning, a habit that runs through Lessons 04, 05 and 08.

The opening of the Soviet Union in the late 1980s made foreign travel possible. In 1990 he was invited to spend three months at the Max Planck Institute for Mathematics in Bonn, directed by Friedrich Hirzebruch.

Bonn and the thesis, 1990–1992

In Bonn, Kontsevich heard Michael Atiyah lecture on a new conjecture of Edward Witten. Witten had argued from two-dimensional quantum gravity that certain intersection numbers on the moduli spaces of algebraic curves are governed by the Korteweg–de Vries equation, a classical equation for water waves. The argument was physical, and it relied on the claimed equivalence of two models of quantum gravity.

Kontsevich found a proof. He built a combinatorial model of the moduli spaces from ribbon graphs, computed the intersection numbers as sums over these graphs, and recognised the sums as the Feynman expansion of a new matrix integral, a matrix version of the Airy function. He then showed that this integral satisfies the KdV hierarchy. Lesson 04 gives the details. The Max Planck Institute extended his stay, and he registered as a doctoral student at the University of Bonn under Don Zagier. His thesis, Intersection theory on the moduli space of curves and the matrix Airy function, was accepted in 1992, and the paper appeared in Communications in Mathematical Physics the same year.

The proof made him famous at once. In 1992 he received the Otto Hahn Medal of the Max Planck Society and one of the prizes awarded by the City of Paris at the first European Congress of Mathematics. He spent 1991–92 visiting Harvard and 1992–93 at the Institute for Advanced Study, and in 1993 he became a professor at Berkeley.

Witten’s conjecture and ribbon graphs

For the general reader. The algebraic curves of a given genus form a geometric space of their own, the moduli space. Witten predicted that certain numbers describing its shape are generated by a solution of a classical wave equation. Kontsevich proved it by cutting each curve into polygons, which turns the problem into counting graphs drawn on surfaces, and then counting those graphs with a matrix integral.

Intersection numbers

Let \(\overline{\mathcal M}_{g,n}\) be the Deligne–Mumford moduli space of stable curves of genus \(g\) with \(n\) marked points, a compact orbifold of complex dimension \(3g-3+n\). The cotangent line at the \(i\)-th marked point forms a line bundle \(\mathbb L_i\), and \(\psi_i=c_1(\mathbb L_i)\). Define

which is zero unless \(d_1+\cdots+d_n=3g-3+n\). Form the generating function \(F(t_0,t_1,\dots)=\sum_g\big\langle\exp\big(\sum_k t_k\tau_k\big)\big\rangle_g\) and set \(U=\partial^2F/\partial t_0^2\).

Kontsevich’s proof

Using Strebel differentials, Kontsevich identified \(\mathcal M_{g,n}\times\mathbb R_{\gt0}^n\) with a space of metric ribbon graphs: graphs with a cyclic order of edges at each vertex, of genus \(g\) with \(n\) labelled boundary cycles, and positive edge lengths. On this cell complex he wrote explicit forms representing the \(\psi_i\) and integrated them over cells. The result is his main identity: for formal variables \(\lambda_1,\dots,\lambda_n\),

where \(G_{g,n}\) is the set of trivalent ribbon graphs of genus \(g\) with \(n\) labelled boundary cycles, and \(\lambda(e),\lambda'(e)\) are the variables of the two boundary cycles on either side of the edge \(e\). The right side is the Feynman expansion of the matrix Airy integral over \(N\times N\) Hermitian matrices,

with \(\Lambda=\operatorname{diag}(\lambda_1,\dots,\lambda_N)\). The final step shows that this integral is a tau-function of the KdV hierarchy, using its expression as a ratio of determinants. Later proofs were given by Andrei Okounkov and Rahul Pandharipande, by Maryam Mirzakhani through volumes of moduli spaces of hyperbolic surfaces, and by others.

The Kontsevich integral

For the general reader. Around 1990 Victor Vassiliev organised knot invariants by a notion of complexity, the invariants of finite type. They seemed to be determined by simple combinatorial data called chord diagrams, but no one could show that every allowed set of data actually comes from an invariant. Kontsevich did, with an explicit integral that produces all of them at once.

Finite type invariants

Extend a knot invariant \(v\) to knots with transverse double points by \(v(K_\times)=v(K_+)-v(K_-)\), where \(K_\pm\) resolve one double point into a positive or negative crossing. Then \(v\) is of type \(\le n\) if it vanishes on every knot with \(n+1\) double points. The weight system of \(v\) is the function it induces on chord diagrams with \(n\) chords, which record the pairing of the double points along the circle. Weight systems satisfy the four-term relation (4T), and those of framing-independent invariants satisfy the one-term relation (1T), which kills any diagram with a chord crossed by no other chord. Let \(\mathcal A\) be the graded vector space of chord diagrams modulo these relations.

Place the knot in \(\mathbb C\times\mathbb R\) with height function \(t\), in Morse position. The Kontsevich integral is

where on each level \(t_j\) a pair \(P\) chooses two points \(z_j,z_j'\) of the knot, \(D_P\) is the chord diagram formed by joining the chosen pairs, and \(\downarrow_P\) counts the chosen points on downward-oriented strands. Invariance under horizontal moves follows from the Knizhnik–Zamolodchikov connection being flat. Moving a critical point changes \(Z\) by a universal factor, which is removed by dividing by a power of \(Z\) of a single hump: \(I(K)=Z(K)/Z(\text{hump})^{c/2}\), with \(c\) the number of critical points. Rationality of the coefficients follows from the existence of rational Drinfeld associators.

Stable maps and counting curves

For the general reader. Through two points passes one line; through five points, one conic. How many curves of degree four that can be parametrized by a single complex variable pass through eleven given points? Such questions had resisted algebraic geometers for a century. In 1994 Kontsevich, with Yuri Manin, answered them all at once with a recursion drawn from string theory.

Stable maps

For a smooth projective variety \(X\) and a class \(\beta\in H_2(X,\mathbb Z)\), a stable map of genus 0 with \(n\) marked points is a map \(f:C\to X\) from a connected nodal curve of arithmetic genus 0 with \(f_*[C]=\beta\), such that each component contracted by \(f\) carries at least three special points. Kontsevich introduced the moduli space \(\overline{\mathcal M}_{0,n}(X,\beta)\) of such maps. It is compact. For homogeneous spaces such as \(\mathbb P^r\) it is an orbifold of the expected dimension \(\dim X+\int_\beta c_1(TX)+n-3\). The Gromov–Witten invariants are integrals \(\int\prod_i\mathrm{ev}_i^*\gamma_i\) of classes pulled back from \(X\) by evaluation at the marked points. They define the quantum product on \(H^*(X)\), and associativity of that product (the WDVV equations) gives relations among them.

Proof idea. Consider rational curves of degree \(d\) through \(3d-2\) general points, two of which, \(p_1,p_2\), are joined to two lines \(\ell_1,\ell_2\): require the curve to pass through \(p_1,p_2\) and to meet \(\ell_1,\ell_2\). Map the moduli space of such curves with these four marked points to \(\overline{\mathcal M}_{0,4}\cong\mathbb P^1\). The fibres over the two boundary points \((p_1p_2\,|\,\ell_1\ell_2)\) and \((p_1\ell_1\,|\,p_2\ell_2)\) are linearly equivalent, so they have the same number of points. Each fibre consists of reducible curves \(C_1\cup C_2\) of degrees \(d_1+d_2=d\). Counting how the points are distributed between the two components, and the \(d_1d_2\) choices of node, gives the recursion.

In 1995 Kontsevich used localization with respect to a torus action on spaces of stable maps to count rational curves on the quintic threefold, checking predictions made by Philip Candelas, Xenia de la Ossa, Paul Green and Linda Parkes from mirror symmetry.

Homological mirror symmetry

For the general reader. String theorists found that Calabi–Yau spaces come in mirror pairs: the complex geometry of one matches the symplectic geometry of the other, and hard curve counts on one side become easy integrals on the other. At the 1994 International Congress, Kontsevich proposed that the true statement is an equivalence of two algebraic structures, one built from coherent sheaves and one from Lagrangian submanifolds. The proposal reorganised the subject.

For a smooth projective variety \(X\), the bounded derived category \(D^b\mathrm{Coh}(X)\) of coherent sheaves is a triangulated category whose morphism spaces are \(\mathrm{Ext}^\bullet(E,F)\). For a symplectic manifold \(Y\), the Fukaya category has as objects Lagrangian submanifolds (with extra data), morphism spaces given by Floer cohomology \(HF^\bullet(L_1,L_2)\), generated by intersection points, and composition counting holomorphic polygons. It is an \(A_\infty\)-category rather than an ordinary one.

Alexander Polishchuk and Eric Zaslow proved the conjecture for elliptic curves in 1998. Paul Seidel proved it for the quartic K3 surface, and Mohammed Abouzaid, Nick Sheridan and others for further cases including Calabi–Yau hypersurfaces in projective space. In 2001 Kontsevich and Yan Soibelman connected the conjecture to the Strominger–Yau–Zaslow picture of mirror symmetry as duality of torus fibrations.

The elliptic curve

Let \(E\) be an elliptic curve and \(E^\vee=\mathbb R^2/\mathbb Z^2\) the mirror symplectic torus. A line bundle of degree \(d\) corresponds to a closed line of slope \(d\) on \(E^\vee\), and more generally a stable bundle of rank \(r\) and degree \(d\) corresponds to a line in homology class \((r,d)\). The structure sheaf corresponds to a horizontal line. Floer cohomology of two transverse lines is generated by their intersection points.

Deformation quantization

For the general reader. Quantum mechanics replaces the commuting functions of classical mechanics by non-commuting operators, with the commutator controlled by Planck’s constant. Can every classical system with a Poisson bracket be deformed into a quantum one this way, at least as a formal power series in Planck’s constant? Kontsevich proved in 1997 that it always can, and gave a universal formula built from graphs.

A Poisson structure on a manifold \(M\) is a bivector \(\pi=\pi^{ij}\partial_i\wedge\partial_j\) whose bracket \(\{f,g\}=\pi^{ij}\partial_if\,\partial_jg\) satisfies the Jacobi identity. A star product is an associative \(\mathbb R[[\hbar]]\)-bilinear product on \(C^\infty(M)[[\hbar]]\) of the form

with each \(B_n\) a bidifferential operator. For a constant Poisson structure on \(\mathbb R^{2n}\), the Moyal product \(f\star g=f\exp\big(\tfrac\hbar2\pi^{ij}\overleftarrow{\partial_i}\overrightarrow{\partial_j}\big)g\) is a star product.

On \(\mathbb R^d\) the formula is explicit:

Here \(G_{n,2}\) is a set of admissible directed graphs with \(n\) internal vertices, each emitting two edges, and two external vertices labelled \(f\) and \(g\). The operator \(B_{\Gamma,\pi}\) places a copy of \(\pi\) at each internal vertex and turns each edge into a partial derivative of its target. The weight \(w_\Gamma\) is an integral over the configuration space of \(n\) points in the upper half-plane of a product of angle forms measured in the hyperbolic metric. Associativity follows from Stokes’ theorem on compactified configuration spaces. Alberto Cattaneo and Giovanni Felder later derived the formula as a perturbative expansion of a two-dimensional topological field theory, the Poisson sigma model.

Motivic integration

For the general reader. Two Calabi–Yau manifolds that agree outside small subsets are called birational. Victor Batyrev proved that they have the same Betti numbers, by counting points over finite fields and integrating over \(p\)-adic numbers. In a 1995 lecture at Orsay, Kontsevich replaced the \(p\)-adic numbers by formal power series and the measure by a value in a ring of varieties. This motivic integration proved that the finer Hodge numbers agree too.

The Grothendieck ring and arcs

The Grothendieck ring \(K_0(\mathrm{Var}_{\mathbb C})\) is generated by classes \([X]\) of varieties, with \([X]=[Z]+[X\setminus Z]\) for closed \(Z\subseteq X\) and \([X\times Y]=[X][Y]\). Write \(\mathbb L=[\mathbb A^1]\). The Hodge–Deligne polynomial \(E(X;u,v)\) defines a ring homomorphism to \(\mathbb Z[u,v]\), with \(E(\mathbb L)=uv\). For a smooth variety \(X\) of dimension \(n\), the arc space \(\mathcal L(X)\) parametrizes morphisms \(\operatorname{Spec}\mathbb C[[t]]\to X\). A cylinder set, determined by the truncation of arcs modulo \(t^{m+1}\), has measure \([\pi_m(A)]\,\mathbb L^{-n(m+1)}\) in a completion \(\hat{\mathcal M}\) of \(K_0(\mathrm{Var})[\mathbb L^{-1}]\).

Proof sketch. Take a common resolution \(Z\to X\), \(Z\to X'\). Because \(K_X\) and \(K_{X'}\) are trivial, the two relative canonical divisors on \(Z\) are equal. Applying the change of variables to both maps gives \(\mu(\mathcal L(X))=\mu(\mathcal L(X'))\), and \(\mu(\mathcal L(X))=[X]\mathbb L^{-n}\). Hence \([X]=[X']\) in \(\hat{\mathcal M}\), and applying \(E\) gives equal Hodge–Deligne polynomials, which for smooth projective varieties determine the Hodge numbers. Jan Denef and François Loeser developed the theory in full and extended it to singular varieties.

Wall-crossing and scattering

For the general reader. In supersymmetric theories, stable particles can suddenly decay or bind together when parameters cross certain walls. The counts of these particles jump, but in a controlled way. In 2008 Kontsevich and Yan Soibelman found the algebraic rule: an ordered product of simple transformations stays the same on both sides of a wall. The same rule, as scattering diagrams, later produced canonical bases for cluster algebras.

Let \(\Gamma\cong\mathbb Z^2\) carry the skew form \(\langle(a,b),(c,d)\rangle=ad-bc\), and write \(x^\gamma\) for the monomial with exponent \(\gamma\), with \(x=x^{(1,0)}\), \(y=x^{(0,1)}\). For \(\gamma\in\Gamma\), define the automorphism of the field \(\mathbb Q(x,y)\)

These preserve the symplectic form \(\frac{dx}{x}\wedge\frac{dy}{y}\).

In the pentagon identity, the left side describes a regime with two stable states of charges \((1,0)\) and \((0,1)\), and the right side the regime after a wall, where a bound state of charge \((1,1)\) has appeared. Davide Gaiotto, Gregory Moore and Andrew Neitzke derived the formula physically for \(N=2\) theories, and Dominic Joyce and Yinan Song developed a parallel theory of generalized Donaldson–Thomas invariants. In the language of Mark Gross and Bernd Siebert, the identity says that the scattering diagram in the hero picture is consistent. Gross, Paul Hacking, Seán Keel and Kontsevich used such diagrams in 2018 to prove the positivity conjecture for cluster algebras and to build canonical bases for them, work recognised by the AMS Moore Prize in 2025.

Berkeley, IHÉS and beyond

After visits to Harvard, the Institute for Advanced Study and Bonn, Kontsevich was a professor at the University of California, Berkeley, from 1993. In 1995 he became a permanent professor at the Institut des Hautes Études Scientifiques at Bures-sur-Yvette, near Paris, the institute where Alexander Grothendieck had built modern algebraic geometry. He has remained there since, holding the AXA–IHÉS Chair from 2008. He became a French citizen in 1999. From 1997 to 2002 he was also a Distinguished Visiting Professor at Rutgers University, and since 2007 he has been a Distinguished Professor at the University of Miami.

His plenary address at the 1994 International Congress in Zürich proposed homological mirror symmetry. At the 1998 Congress in Berlin he received the Fields Medal, together with Richard Borcherds, Timothy Gowers and Curtis McMullen. The Congress press release described contributions to four problems of geometry, among them the proof of Witten’s conjecture and the construction of a universal knot invariant. Clifford Taubes gave the laudation.

His later work includes Periods (2001) with Don Zagier, which defined the class of numbers given by integrals of algebraic forms over algebraic domains and posed conjectures about them; a long collaboration with Yan Soibelman on \(A_\infty\)-categories, non-commutative geometry, mirror symmetry and wall-crossing; work with Alexei Belov-Kanel showing, independently of Yoshifumi Tsuchimoto, that the Jacobian conjecture and the Dixmier conjecture are stably equivalent; and the 2018 paper on cluster algebras with Gross, Hacking and Keel. Since 2019 he has led the European Research Council Synergy project ReNew Quantum on resurgence and exact methods in quantum theory. Many of his ideas first circulated as lectures, preprints and short papers that others then developed.

His doctoral students include Serguei Barannikov. His influence travels mainly through collaborators and through the many researchers who have turned his papers and talks into research programmes.

Honours and influence

Kontsevich’s honours include the Otto Hahn Medal of the Max Planck Society and a prize of the City of Paris at the first European Congress of Mathematics (1992); the Henri Poincaré Prize of the International Association of Mathematical Physics (1997); the Fields Medal (1998); the Crafoord Prize in Mathematics, shared with Edward Witten (2008); the Shaw Prize in Mathematical Sciences (2012), cited for work in algebra, geometry and mathematical physics including deformation quantization, motivic integration and mirror symmetry; the Fundamental Physics Prize (2012); the Breakthrough Prize in Mathematics in its first year (2014); and the E. H. Moore Research Article Prize (2025). He is a member of the French Academy of Sciences (2002), the Academia Europaea (2000) and the Accademia dei Lincei, a Foreign Associate of the US National Academy of Sciences (2015) and a Chevalier of the Légion d’honneur (2004).

Legacy

• Moduli and enumerative geometry: stable maps and Gromov–Witten theory are the standard framework for counting curves, and the ribbon-graph model of moduli space is used across topology and mathematical physics.
• Mirror symmetry: homological mirror symmetry is a central organising conjecture of symplectic and algebraic geometry, linked to Fukaya categories and the SYZ picture.
• Quantization: the formality theorem made operads, \(L_\infty\)-algebras and graph complexes everyday tools.
• Knots and arithmetic: the Kontsevich integral connects knot theory to Drinfeld associators and multiple zeta values, and the theory of periods frames questions in transcendence.
• Wall-crossing: the Kontsevich–Soibelman formula links Donaldson–Thomas theory, supersymmetric field theory and cluster algebras.

Connections

People linked to Maxim Kontsevich across this collection. A name without a link has no page here yet. See Maxim Kontsevich in the network of all 337 lives →

Teachers and advisors

• Don Zagier 1992

Students

• Serguei Barannikov

Influenced by

• Edward Witten
• Israel Gelfand
• Michael Atiyah
• Vladimir Drinfeld
• Victor Vassiliev

Influenced

• Maryam Mirzakhani
• Dror Bar-Natan
• François Loeser
• Jan Denef

Colleagues and collaborators

• Alex Eskin
• Alexander Kirillov 1983
• Alexei Belov-Kanel
• Friedrich Hirzebruch 1990-1992
• Mark Gross 2014-2018
• Paul Hacking 2014-2018
• Seán Keel 2014-2018
• Yan Soibelman
• Yuri Manin 1994

Also mentioned in this life

• George Biddell Airy
• Pierre Deligne
• Leonhard Euler
• Mikhail Gromov
• Alexander Grothendieck
• Otto Hahn
• Werner Heisenberg
• Carl Gustav Jacob Jacobi
• Sophus Lie
• Curtis T. McMullen
• David Mumford
• Max Planck
• Henri Poincaré
• Siméon Denis Poisson
• Bernhard Riemann
• George Gabriel Stokes
• Hermann Weyl

Fields

• algebraic geometry 17 lives
• mathematical physics 45 lives
• knot theory 3 lives

Sources and further reading

Biography

1. J. J. O’Connor and E. F. Robertson, Maxim Lvovich Kontsevich, MacTutor History of Mathematics.
2. Maxim Kontsevich, Wikipedia.
3. IHÉS, Maxim Kontsevich, mathematician.
4. Academia Europaea, Maxim Kontsevich.
5. Aarhus University, Honorary doctorate: Professor Maxim Kontsevich.
6. IHÉS, Maxim Kontsevich, recipient of the 2025 AMS Moore Prize.
7. C. H. Taubes, The work of Maxim Kontsevich, Documenta Mathematica, Extra Volume ICM 1998, vol. I.

Mathematics

1. E. Witten, Two-dimensional gravity and intersection theory on moduli space, Surveys in Differential Geometry 1 (1991) 243–310.
2. M. Kontsevich, Intersection theory on the moduli space of curves and the matrix Airy function, Comm. Math. Phys. 147 (1992) 1–23.
3. M. Kontsevich, Vassiliev’s knot invariants, Adv. Soviet Math. 16 (1993) 137–150.
4. D. Bar-Natan, On the Vassiliev knot invariants, Topology 34 (1995) 423–472.
5. M. Kontsevich and Yu. Manin, Gromov–Witten classes, quantum cohomology, and enumerative geometry, Comm. Math. Phys. 164 (1994).
6. M. Kontsevich, Homological algebra of mirror symmetry, Proceedings of the ICM, Zürich 1994.
7. M. Kontsevich, Enumeration of rational curves via torus actions, in The Moduli Space of Curves, Birkhäuser, 1995.
8. M. Kontsevich, Deformation quantization of Poisson manifolds, Lett. Math. Phys. 66 (2003); preprint 1997.
9. J. Denef and F. Loeser, Germs of arcs on singular algebraic varieties and motivic integration, Invent. Math. 135 (1999) 201–232.
10. A. Polishchuk and E. Zaslow, Categorical mirror symmetry: the elliptic curve, Adv. Theor. Math. Phys. 2 (1998).
11. M. Kontsevich and D. Zagier, Periods, in Mathematics Unlimited: 2001 and Beyond, Springer, 2001.
12. M. Kontsevich and Y. Soibelman, Stability structures, motivic Donaldson–Thomas invariants and cluster transformations, arXiv:0811.2435 (2008).
13. M. Gross, P. Hacking, S. Keel and M. Kontsevich, Canonical bases for cluster algebras, J. Amer. Math. Soc. 31 (2018).