<sod>In 2016, [[Maryna Viazovska]] announced a proof that the [[E8 lattice]] provides the optimal packing (regardless of regularity) in eight-dimensional space, and soon afterwards she and a group of collaborators announced a similar proof that the [[Leech lattice]] is optimal in 24 dimensions. The proofs involve using the [[Laplace transform]] of a carefully-chosen [[modular form]] to construct a [[radial symmetry|radially-symmetric]] function <math>f</math> such that <math>f</math> and its [[Fourier transform]] <math>\hat f</math> both equal one at the [[origin (mathematics)|origin]], and both vanish at all other points of the optimal lattice, with <math>f</math> negative outside the central sphere of the packing and <math>\hat f</math> positive. Then, the [[Poisson summation formula]] for <math>f</math> is used to compare the density of the optimal lattice with that of any other packing.<ref>{{citation|url=https://www.youtube.com/watch?v=8qlZjarkS_g|first=Stephen D.|last=Miller|date=April 4, 2016|title=The solution to the sphere packing problem in 24 dimensions via modular forms|publisher=[[Institute for Advanced Study]]}}. Video of an hour-long talk by one of Viazovska's co-authors explaining the new proofs.</ref> Although it has not yet been refereed and published, mathematician [[Peter Sarnak]] called the proof "stunningly simple" and wrote that "You just start reading the paper and you know this is correct."<ref>{{citation|last1=Klarreich|first1=Erica|authorlink1=Erica Klarreich|title=Sphere Packing Solved in Higher Dimensions|url=https://www.quantamagazine.org/20160330-sphere-packing-solved-in-higher-dimensions|magazine=Quanta Magazine|date=March 30, 2016}}</ref><eod>
<sod>In 2016, [[Maryna Viazovska]] announced a proof that the [[E8 lattice]] provides the optimal packing (regardless of regularity) in eight-dimensional space, and soon afterwards she and a group of collaborators announced a similar proof that the [[Leech lattice]] is optimal in 24 dimensions. Both of these lattices were previously known to be the densest lattice packings in these dimensions.<ref>{{Citation |  last1=Cohn | first1=Henry | last2=Kumar | first2=Abhinav | title=Optimality and uniqueness of the Leech lattice among lattices | doi=10.4007/annals.2009.170.1003 | mr=2600869 | zbl=1213.11144 | year=2009 | journal=Annals of Mathematics | issn=1939-8980 | volume=170 | issue=3 | pages=1003–1050 | arxiv=math.MG/0403263 }} {{Citation | last1=Cohn | first1=Henry | last2=Kumar | first2=Abhinav | title=The densest lattice in twenty-four dimensions | doi=10.1090/S1079-6762-04-00130-1  | mr=2075897  | year=2004 | journal=Electronic Research Announcements of the American Mathematical Society | issn=1079-6762 | volume=10 | issue=07 | pages=58–67  |arxiv=math.MG/0408174 }}</ref><eod>