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In this research announcement we outline the methods used in our recent proof that the Leech lattice is the unique densest lattice in $\mathbb{R}^{24}$. Complete details will appear elsewhere, but here we illustrate our techniques by applying them to the case of lattice packings in $\mathbb{R}^2$, and we discuss the obstacles that arise in higher dimensions.

Copyright 2004 American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

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Article Info

• ERA Amer. Math. Soc. 10 (2004), pp. 58-67
• Publisher Identifier: S 1079-6762(04)00130-1
• 2000 Mathematics Subject Classification. Primary 11H31, 52C15; Secondary 05B40, 11H55
• Key words and phrases.
• Received by editors April 14, 2004
• Posted on June 17, 2004
• Communicated by Brian Conrey
• Comments (When Available)

Henry Cohn

Microsoft Research, One Microsoft Way, Redmond, WA 98052-6399

E-mail address: cohn@microsoft.com

Abhinav Kumar

Department of Mathematics, Harvard University, Cambridge, MA 02138

E-mail address: abhinav@math.harvard.edu

Kumar was supported by a summer internship in the Theory Group at Microsoft Research.