Maxim Kontsevich, recipient of the 2025 AMS Moore Prize

IHES is pleased to announce that Maxim Kontsevich has been awarded the 2025 Moore Prize by the American Mathematical Society (AMS).

This prize, established in 2004 and named in honor of E.H. Moore, a former AMS president, is awarded every three years. It recognizes a research article published in one of the AMS journals, including the Journal of the AMS, Proceedings of the AMS, Transactions of the AMS, AMS Memoirs, Mathematics of Computation, the Electronic Journal of Conformal Geometry and Dynamics, and the Electronic Journal of Representation Theory.

For the 2025 edition of the prize, Maxim Kontsevich and his co-authors Mark Gross (University of Cambridge), Paul Hacking (University of Massachusetts Amherst), and Seán Keel (University of Texas at Austin) have been distinguished for their article Canonical Bases for Cluster Algebras, published in the Journal of the American Mathematical Society in 2018.

Bernhard Keller, Professor of Mathematics at Université Paris Cité, reflects on this groundbreaking article:

“Cluster algebras, invented by Fomin-Zelevinsky in 2002, are certain commutative algebras endowed with a rich combinatorial structure. Among these algebras are the homogeneous coordinate algebras of the Grassmannian, of the double Bruhat cells and of many other varieties of importance in geometry and Lie theory. Fomin-Zelevinsky’s main hope was to obtain a combinatorial approach to the construction of Lusztig’s “canonical bases”, which such coordinate algebras possess, and to his closely related theory of total positivity.

Fock-Goncharov made Fomin-Zelevinsky’s hope more precise in their famous duality conjectures but even using this framework, it turned out to be extremely hard to construct “canonical bases” for cluster algebras in the desired generality. Similarly, Fomin-Zelevinsky’s positivity conjecture from 2002 remained almost completely open until 2013, when Lee-Schiffler proved it for cluster algebras arising from quivers. In this paper, Gross-Hacking-Keel-Kontsevich make two breakthroughs:

• They show the positivity conjecture in full generality for cluster algebras arising from arbitrary valued quivers;
  
• They construct “canonical bases” for large classes of cluster algebras and in particular for all the examples from Lie theory and higher Teichmüller theory, thus (partially) confirming Fock-Goncharov’s conjectures.

They obtain these results using tools from mirror symmetry (scattering diagrams and broken lines), developed notably by Kontsevich-Soibelman and Gross-Siebert, whose applicability to cluster algebras was a deep insight of Gross, Hacking and Keel. Through both, its results and its methods, the paper has been transformative for the study of cluster algebras, with important repercussions in the large array of other fields where they have been shown to be relevant, in particular enumerative geometry, representation theory, and quantum topology.”

IHES warmly congratulates Mark Gross, Paul Hacking, and Seán Keel—all former visitors to the Institute—as well as Maxim Kontsevich on receiving this prestigious prize.

Read the official announcement of the Moore Prize here.

Read Seán Keel’s background story on [GHKK] here.