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Paul Hacking, professor in the Department of Mathematics and Statistics in the College of Natural Sciences, has been named one of four co-recipients of the 2025 E.H. Moore Research Article Prize by the American Mathematical Society.

The E.H. Moore Research Article Prize is awarded every three years for an outstanding research article that appeared in one of the AMS’s primary research journals during the six calendar years ending a full year before the meeting at which the prize is awarded.

Hacking and his co-authors – Mark Gross of the University of Cambridge, U.K., Seán Keel of the University of Texas at Austin, and Maxim Kontsevich of the Institut des Hautes Études Scientifiques, France – have been honored for their paper, “Canonical Bases for Cluster Algebras,” which was published in the April 2018 edition of the Journal of the American Mathematical Society.

Their paper “introduced new important constructions and techniques on cluster algebras,” the AMS says. “Using the combinatorics of Gross-Siebert scattering diagrams and broken lines, they defined the ‘theta series’ associated with an arbitrary g-vector; this enabled them to produce, for the first time, an explicit construction of a ‘canonical’ basis in a cluster algebra.”

The award citation notes that “by introducing novel mirror symmetry techniques into the field, Gross, Hacking, Keel, and Kontsevich accomplished several breakthroughs on the main problems of general structure theory of cluster algebras and varieties. Among other things, they solved the Laurent positivity conjecture of Fomin and Zelevinsky and the duality conjecture of Fock and Goncharov.”

“I’d like to thank the AMS for their kind recognition of our work,” Hacking said in response to winning the prize. “I’d also like to acknowledge the dependence of our work on prior work of the following researchers: Sergey Fomin and Andrei Zelevinsky, who developed the theory of cluster algebras around 2001; Vladimir Fock and Alexander Goncharov, who recast that theory in a more geometric form in 2003 and formulated an intriguing duality conjecture; Kontsevich-Soibelman and Gross-Siebert, who developed the notion of scattering diagrams in mirror symmetry (building on work of Fukaya) which plays a crucial role in our paper, allowing us to prove the Fock-Goncharov duality conjecture and construct canonical bases of cluster algebras as envisaged by Fomin-Zelevinsky. I’d like to thank the cluster algebra community for their openness to assimilating new ideas and kindness in patiently explaining existing theory to us. Thanks to IHES for hosting us during some of our work on this project. Finally, thanks to my family for their love and support.”

Hacking and his co-authors will be presented the $5,000 prize at the 2025 Joint Mathematics Meetings Jan. 8-11 in Seattle. Their winning article, “Canonical Bases for Cluster Algebras,” can be found online from the Journal of the American Mathematical Society, and the complete announcement of the Moore Research Article Prize can be found on the AMS website.