184 NAW 5/17 nr. 3 september 2016 A breakthrough in : the search for magic functions David de Laat, Frank Vallentin

David de Laat Frank Vallentin Centrum Wiskunde & Informatica Mathematisches Institut Amsterdam Universität zu Köln, Germany [email protected] [email protected]

A breakthrough in sphere packing: the search for magic functions

This paper by David de Laat and Frank Vallentin is an exposition about the two recent she announced her spectacular result in the breakthrough results in the theory of sphere packings. It includes an interview with Henry paper titled ‘The sphere packing problem Cohn, Abhinav Kumar, Stephen D. Miller and . in 8’ [10] on the arXiv-preprint server. Only one week later, on 21 March The sphere packing problem asks for a in three is the fact that there 2016, Henry Cohn, Abhinav Kumar, Stephen densest packing of congruent solid spheres are uncountably many inequivalent optimal D. Miller, Danylo Radchenko and Maryna Vi- in n-dimensional space Rn. In a packing packings. In 2014 a fully computer verified azovska announced a proof for the n = 24 the (solid) spheres are allowed to touch on version of Hales’ proof was completed; it case [3], building on Viazovska’s work. their boundaries, but their interiors should was a result of the collaborative Flyspeck Here we want to illustrate that the opti- not intersect. project, also directed by Hales [7]. mal sphere packings in dimensions 8 and While the case of the real line, n = 1, Recently, Maryna Viazovska, a postdoc- 24 are very special (in the next section we is trivial, the case n = 2 of packing circles toral researcher from working at the give constructions of the E8 and of in the plane was first solved in 1892 by Humboldt University of Berlin, solved the the K24, which provide the the Norwegian mathematician Thue (1863– eight-dimensional case. On 14 March 2016 optimal sphere packings in their dimen- 1922). He showed that the honeycomb sions), and we aim to explain the main hexagonal lattice gives an optimal packing; ideas of the recent breakthrough results in see Figure 1. sphere packing: The first rigorous proof is due to the Hungarian mathematician Fejes Tóth (1915– Theorem 1. The lattice E8 is the densest 8 2005) in 1940. He also proved that this packing in R . The Leech lattice K24 is packing is unique (up to rotations, transla- the densest packing in R24. Moreover, no tions, and uniform scaling) among period- other periodic packing achieves the same ic packings. For n = 3, the sphere packing density in the corresponding dimension. problem is known as the . It was solved by the American mathemati- In the last section we will see that cian Hales in 1998 following an approach the beautiful proofs of these theorems by Fejes Tóth. Hales’ proof is extremely use ideas from analytic number theory. complex, takes more than 300 pages, and Viazovska found a ‘magic’ function for di- makes heavy use of computers. One of the mension 8, which together with the linear Figure 1 The hexagonal lattice and the corresponding difficulties of the sphere packing problem circle packing. programming bound of Cohn and Elkies, David de Laat, Frank Vallentin A breakthrough in sphere packing: the search for magic functions NAW 5/17 nr. 3 september 2016 185

as explained later on, gives a proof for the The optimality of the E8 lattice. Her method The nicest lattices are those which are gave a hint how to find a magic function even and unimodular. However, they only for dimension 24. Although the proof is rel- occur in higher dimensions: one can show atively easy to understand, and basically that the first appearance of such an even no computer assistance is needed for its and is in dimension 8. verification, computer assistance was cru- It is the E8 lattice, which was first explic- cial to conjecture the existence of, and to itly constructed by the Russian mathema- find, these magic functions. ticians Korkine (1837–1908) and Zolotareff (1847–1878) in 1873.

Optimal lattices Here we give a construction of the E8 In this section we introduce the two ex- Figure 2 A periodic packing that is not a lattice packing. lattice which is based on lifting binary er- ceptional sphere packings in dimension 8 ror correcting codes. For this we define the and 24. The book Sphere Packings, Lattic- (extended) Hamming code H8 via a regu- n es, and Groups of Conway and Sloane [5] vol(R /)Lb= |(det 1,,f bn)|. lar three-dimensional tetrahedron: consid- is the definitive reference on this topic; er the binary linear code H , which is the The density of L is then given by 8 the Italian-American combinatorialist Rota vector space over the finite fieldF 2 (con- (1932–1999) reviewed the book saying: sisting of the elements and ) spanned vol(Brn (/1 2)) 0 1 D()L = n , by the rows of the matrix “This is the best survey of the best vol(R /)L work in one of the best fields of combi- J1000 0111N where r1 is the shortest nonzero vector K O natorics, written by the best people. It K O length in L. Here, Brn (/1 2) is the solid K0100 1011O 48# GI==(|A),K O ! F2 will make the best reading by the best sphere of radius r1 /2, whose volume is K0010 1101O students interested in the best mathe- K0001 1110O L P rn/2 matics that is now going on.’’ vol(Br(/22)) = (/r )n , where I is the identity matrix and where A n 11C(/n 21+ ) is the adjacency matrix of the vertex-edge Lattice packings where C is the gamma function; it satisfies graph of a three-dimensional tetrahedron How does one define a packing of unit the equation CC()xx+=1 ()x , and two with vertices vv12,,vv34, . Hence, the Ham- spheres in n-dimensional space? In gener- particularly useful values are C()11= and ming code is a 4-dimensional subspace of al, such a packing is defined by the set of 8 4 C(/12) = r. the vector space F2 . It consists of 21= 6 centers L of the spheres in the packing. The optimal sphere packing density code words: We talk about lattice packings when L can be approached arbitrarily well by the 0000 ||0000 1000 0111 1100 | 1100 0111 | 1000 1111 | 1111 forms a lattice. Then there are n linearly density of a periodic packing. In a periodic 0100 | 1011 1010 | 1010 1011 | 0100 independent vectors bb1,,f n ! L, called a packing the set of centers is the union of a 0010 | 1101 1001 | 1001 1101 | 0010 lattice basis of L, so that L is the set of 0001 | 1110 0110 | 0110 1110 | 0001 finite numberm of translates of a lattice L: 0101 | 0101 integral linear combinations of bb1,,f n. For 0011 | 0011 {:xv++vL!!}{, xv:}vL instance, the three lattices 12 It is interesting to look at the occurring ! ,,g {:xvm + vL}; Hamming weights (the number of non-zero 2 -1 ZZ,,+ Z see Figure 2 for an example where m = 3. entries) of code words. In H , one code ee0oo3 8 The density of a periodic packing is word has Hamming weight 0, 14 code KJ- 2ON JK 2 NO JK 0 ON n K O K O K O mB$ vol( n ()rL)/vol(R /), where r is the words have Hamming weight 4, and one ZZZK O K O K O K-+2O K-+2O K 2 O radius of the spheres in the packing. It is code word has Hamming weight 8. Since K 0 O K 0 O K- 2O L P L P L P possible that in some dimensions the op- all occurring Hamming weights are divisi- define densest sphere packings in dimen- timal sphere packing is not a lattice pack- ble by four, and four is two times two, the sions 1, 2, and 3. ing; for example, the best known sphere Hamming code H8 is called doubly even. 10 We should also define what we mean packing in R is a periodic packing but it Let us compute the dual code when we talk about density. Intuitively, the is not a lattice packing. 8 9 8 density of a sphere packing is the fraction A few last definitions: From a latticeL H = yx! F2 :(/ i yi = 02mod ) 8 * of space covered by the spheres of the we can construct its dual lattice by i = 1 packing. When the sphere packing is a lat- ! . Ly* = {:!!RZn xy$ forall xL! }. forall x H8 tice, this intuition is easy to make precise: 4 The density of the sphere packing deter- It is not difficult to see that the volume of Squaring the matrix A yields mined by L is the volume of one sphere a lattice and its dual are reciprocal values, 4 nn* 2 divided by the volume of L, that is, the so that vol(RR/)LL$ vol( /)= 1 holds. AAij = / ik Akj volume of a fundamental domain of L. One When a lattice equals its dual (LL= *) and k = 1 ++ possible fundamental domain of L is given when the square of every occurring vector = |{kv:}ikvvand kjv | 3 if ij= , by the parallelepiped spanned by the lat- length is an even , then L is called = 2 if ij! . tice basis bb1,,f n, so that we have an even and unimodular lattice. ) 186 NAW 5/17 nr. 3 september 2016 A breakthrough in sphere packing: the search for magic functions David de Laat, Frank Vallentin

Hence, AI2 = mod 2. From this, GGT =+IA2 this conjecture has been proved in the This is the Leech lattice. It is an even un- = 02mod follows. Hence, we have the in- breakthrough work of Maryna Viazovska. imodular lattice. In K24 there are 196560 9 clusion HH8 3 8 and by considering di- shortest vectors which have length 4. mensions we see that H8 is a self-dual The Leech lattice The occurring vector lengths in K24 are 9 code; that is, HH8 = 8 holds. We turn to 24 dimensions and to the Leech 04,,,,68f We can define the latticeE 8 by the fol- lattice. In 1965 Leech (1926–1992) realized In 1969 Conway showed that the Leech lowing lifting construction (which is usually that he constructed a surprisingly dense lattice again has a remarkable number called Construction A): sphere packing in dimension 24. For his theoretical property: It is the only even construction, he used the (extended bina- unimodular lattice in dimension 24 which 1 8 E = xx:,!!Z x mod 2 H8 . ry) Golay code which is an exceptional er- does not have vectors of length 2. He 8 (22 ror correcting code found by Golay (1902– used this result to determine the auto- Now it is immediate to see that E8 has 240 1989) in 1949. To define the Leech lattice morphism group (the group of orthogonal shortest (nonzero) vectors: we modify the lifting construction of the E8 transformation which leave K24 invariant) 16 = 24 vectors: lattice. We replace the Hamming code by of the Leech lattice and it turned out the the Golay code and apply two extra twists. number of automorphisms equals ! 21eii,,= f,8 For defining the Golay code we replace 224 = 214 $ 4 : |Aut()K |= 2322 $$9457$ 2 $$$11 13 23 vectors the regular tetrahedron in the construction 24 8 1 of the Hamming code by the regular icosa- = 8315553613086720000, / !xi exi,(! H8 and wt x),= 4 2 i = 1^h hedron and we apply the first twist. Con- and that this group contained three new where ee18,,f are the standard basis vec- sider the binary code G24 spanned by the sporadic simple groups Co12,Co.Co3. The 8 tors of R and where wt()xi= |{ :}xi ! 0 | rows of the matrix classification theorem of finite simple denotes the Hamming weight of x. The groups, which was announced in 1980, GI=-(|JA) shortest nonzero vectors of E8 have length says that there are only 26 finite simple KJ100000000000 100000111111ON 2. The occurring vector lengths in E8 are K010000000000 010110001111O sporadic groups. They are sporadic in the K O 02,,,,46f, so that E is an even lat- K001000000000 001011100111O sense that they are not contained in the 8 K O tice. K000100000000 010101110011O infinite families of cyclic groups of prime K000010000000 011010111001O From the lifting construction it follows K O order, alternating groups and groups of Lie K000001000000 001101011101O 12 # 24 = K000000100000 101110101100O ! F2 , that the density of E8 is ||H4 = 16 times K O type. K000000010000 100111010110O the density of the lattice 2 Z8 which is K O Similar to the eight-dimensional case, K000000001000 110011101010O spanned by 22ee18,,f . Thus, K000000000100 111001110100O by results of Odlyzko, Sloane, Levenshtein, K000000000010 111100011010O Bannai and Sloane, the 196560 shortest 1 K000000000001 111111000001O vol(RR8 /)= $ vol( 88/)2 Z E8 16 L P vectors of K24 give the unique solution of 1 where we use JA- instead of A, with J the problem in dimension ==$ ()218 , 16 the all-ones matrix and A the adjacen- 24. In 2004 Cohn and Kumar proved the so that E8 is unimodular. One can show cy matrix of the vertex-edge graph of a optimality of the sphere packing of the that E8 is the only even unimodular lattice three-dimensional icosahedron. This code, Leech lattice among lattice packings by a in dimension 8. In general, the lifting con- the extended binary Golay code G24, is a computer assisted proof, see [2] and sec- 24 struction always yields an even unimodular 12-dimensional subspace in F2 . It con- tion ‘Producing numerical evidence’ further lattice when we start with a binary code tains one vector of Hamming weight 0, 759 on. However, despite all the similarities of which is doubly even and self-dual. vectors of Hamming weight 8, 2576 vec- E8 and K24, there is a puzzling difference Next to this exceptional number theo- tors of Hamming weight 16, 759 vectors between E8 and K24 when it comes to retical property, E8 also has exceptional of Hamming weight 20, and one vector of sphere coverings: Schürmann and Vallen- geometric properties: In 1979, Odlyzko and Hamming weight 24; G24 is a doubly even tin [9] showed in 2006 that K24 provides Sloane, and independently Levenshtein, and self-dual code. at least a locally thinnest sphere covering proved that one cannot arrange more vec- We define the even unimodular lattice in the space of 24-dimensional lattices, tors on a sphere in dimension 8 of radius whereas Dutour-Sikirić, Schürmann and 1 24 2 so that the distance between any two L = xx:,!!Z x mod 2 G24 . Vallentin [9] showed in 2012 that one can 24 (22 distinct vectors is also at least 2; the improve the sphere covering of the E8 lat- 240 vectors give the unique solution of Since the minimal non-zero Hamming tice when picking a generic direction in the the kissing number problem in dimension weight occurring in the Golay code is 8, space of eight-dimensional lattices. 8 as was shown in by Bannai and Sloane this lattice has 48 shortest vectors ! 2 ei, in 1981. Blichfeldt (1873–1945) showed with i = 12,,f 4, of length 2. To eliminate Theta series and modular forms them we make the second twist, we define in 1935 that E8 gives the densest sphere As already indicated, the class of even un- packing among lattice packings. For a long 24 imodular lattices is restrictive, at least in time it has been conjectured that E also K24 = xL! 24 : 20/ xi = mod 4 small dimensions. One can show that they 8 *4 gives the unique densest sphere packing i=1 only exist in dimensions that are divisible 24 in dimension 8, without imposing the (se- , (,11f,)+=xx:,! Lx24 22/ i mod 4 . by 8, furthermore for every such dimension vere) restriction to lattice packings. Now * i=1 4 n there are only finitely many even uni- David de Laat, Frank Vallentin A breakthrough in sphere packing: the search for magic functions NAW 5/17 nr. 3 september 2016 187

3 a b 2 2rinz h8 = 1 Mordell, 1938 az + b Ez()=+1 v ()ne , z = . kkg / - 1 eoc d cz + d ()1 - k n = 1 h16 = 2 Witt, 1941 v k - 1 The action of the generator S corresponds where k - 1 ()nd= /dn| is the divisor h24 = 24 Niemeier, 1973 to the involution zz7- 1/ and the action function. We can express the theta series h $ 1162109024 King, 2003 32 of the generator T corresponds to the of the E8 lattice and the Leech lattice K24 Table 1 translation zz7 + 1. by E4 and E6. For the E8 lattice we have A of weight k is a holo- 3 jvr morphic function f|HC" that satisfies E8 ()zE==43()zr1+240 / ()q modular lattices, this number is denoted r =1 the transformation law by hn. In Table 1 we summarize the known =+1 240$$1 qq++240$$9 23240$$28 q +g values of hn. az + b 23 f =+()cz dfk ()z =+1 240$$qq++2160 6720$q +g. A major tool for studying even unimod- b cz + d l ular lattice are their theta series (first stud- a b For the Leech lattice we have forall !!SL2 ()ZH,,z ied by Jacobi (1804–1851)): The theta series ec do 3 jK ()zE=-4 ()zz720D() of a lattice L is 24 and which has a power series expansion =+1 196560 $$qq23++16773120 g 3 in qe= 2riz. Next to theta series, Eisenstein j = nr()qr where D is defined as LL/ series, due to Eisenstein (1823–1852), form r = 0 3 2 ! $ an important class of modular forms. For Ez4 () - Ez6 () with nrL ()=={:xLxx 2r}, D()z = . an integer k $ 3 define 1728 the generating function of the number of 11 When k = 2, we can still write down the lattice vectors of length 2r. In order to Ez()= , k 2g()k / ()cz + d k (1) series as in (1), but then we loose some 2riz (,cd)\! Z2 {}0 work with them analytically we set qe= pleasant properties. For example, the series where z lies in the complex upper half where g is the Riemann zeta function no longer converges absolutely, so the or- 3 1 plane HC= {:zz! Im > 0}, so that jL is g ()s = . For even k $ 3, the /n = 1 s dering of summating matters. Then a function of . The is pe- n z Eisenstein series Ek is a modular form of 11 riodic mod Z: we have jjLL()zz=+()1 . weight k. Ez()= 2 22g()/ / ()cd+ z 2 The Poisson summation formula states Curiously, a theorem of Siegel (1896– c ! Z d ! Z 3 1981) gives a relation between the theta =-124 v ()ne2rinz, 1 2rix $ y / 1 fx()+=v ef()y , series of even unimodular lattices and Ei- n = 1 //vol(Rn /)L V xL! yL! * senstein series. Let LL1,,f hn be the set where we of course omit the pair with v ! Rn, where of even unimodular lattices in dimension (,cd)(= 00,) in the first sum. This for- n. Define bidden Eisenstein series is not a modu- r fy()= fx()ed-2 iy $ x x 11lar form; instead it satisfies the following V # Mn()=+g + . n transformation law R |Aut()LL1 ||Aut( hn)| is the n-dimensional Fourier transform. Then 2 6iz Ez2 (/-=1 )(zE2 z).- r Using the Poisson summation formula one h 11n It is a quasi-modular form. can show that jL satisfies the transforma- Ez()= j ()z . n/2 Mn()/ |Aut()L | Lj tion law j = 1 j The LP Bound of Cohn and Elkies n/2 1 Another striking fact is that one can show jj(/-=1 zz)(/)i * ()z , We can use optimization techniques, in L vol(Rn /)L L that the modular forms form an algebra particular linear and semidefinite program- which is isomorphic to the polynomial al- which in particular shows that jL is a mod- ming, to obtain upper bounds on the opti- gebra C[,EE46]. ular form of weight n/2. From this it is not mal sphere packing density. When k is even, the Eisenstein series difficult to derive that an even unimodular Let us recall some facts about linear has the Fourier expansion lattice can only exist when n is a multiple programming. In a linear program we of 8. want to maximize a linear functional over What is a modular form? The group a polyhedron. For example, we maximize the functional ac7 $ a over all (entrywise) 1 a b T − I T nonnegative vectors a ! Rd satisfying the SL2 ()ZZ=-:,abcd,, ! ,,ad bc = 1 )3eoc d i linear system Aa = b, or we maximize the

2 functional over all (nonnegative) vectors a which is generated by the matrices (ST) S TS satisfying the inequality Aa # b. Linear pro- STS ST ST 1 TST − grams can be solved efficiently in practice 0 -1 1 1 ST= and = by a simplex algorithm (which traverses ee1 0 oo0 1 3/2 1 1/2 0 1/2 1 3/2 − − − a path along vertices of the polyhedron) acts on upper half plane H by fractional or by Karmakar’s interior point method, Figure 3 A fundamental domain of the action of SL2 ()Z linear transformations on the upper half plane H. where the latter runs in polynomial time. 188 NAW 5/17 nr. 3 september 2016 A breakthrough in sphere packing: the search for magic functions David de Laat, Frank Vallentin

The following theorem, the linear pro- where ~ is the normalized invariant mea- given a scalar t ! (,- 11), we seek a larg- gramming bound of Cohn and Elkies sure on Sn - 1, also satisfies these condi- est subset of the sphere Snn- 1 3 R such from 2003, can be used to obtain upper tions. For f|RRn " radial, we (ab)use that the inner product between any two bounds on the sphere packing density. In the notation fr() for the common value distinct points is at least t. A spherical code the statement of this theorem we restrict of f on the vectors of length r. The (in- corresponds to a spherical cap packing to Schwartz functions because the proof, verse) Fourier transform maps radial func- where we center spherical caps of angle which we give here as it is simple and in- tions to radial functions. Moreover, the arccos(/t 2) about the points in the code. 2 sightful, uses the Poisson summation for- Gaussian xe7 -r x is fixed under the The Cohn–Elkies bound can be seen as a mula. A function f|RRn " is a Schwartz Fourier transform, and, more generally, noncompact analogue of a similar bound function if all its partial derivatives exist the sets for the spherical code problem known as and tend to zero faster than any inverse the Delsarte linear programming bound. 2 -r x 2 power of x. There are alternative proofs Pxd = {(7 p x ):e However, because of noncompactness this that do not use Poisson summation and p is a polynomialofdegreeatmostd}. sampling approach does not work well for for which the Schwartz condition can be are invariant under the Fourier transform. the sphere packing problem. weakened. A computer-assisted approach to find good Another approach, based on semidef- functions for the above theorem is to re- inite programming, does work well for Theorem 2. If f|RRn " is a Schwartz noncompact problems such as the sphere strict to functions from Pd for some fixed function and r1 is a positive number value of d. Any function from this set that packing problem. In [8] this approach is with f()01= , fu()$ 0 for all u, and used to compute upper bounds for pack- V V satisfies f()01= can be written as fx()# 0 for x $ r1, then the densi- ings of spheres and spherical caps of sev- n d ty of a sphere packing in R is at most -k n/21- 2 eral radii. Semidefinite programming is a fxa ()= 1 + / akk !(r Lk r x ) fB()02$ vol( n (/r1 )). fpk = 1 powerful generalization of linear program- 2 ming, where we maximize a linear func- # e-r x (2) Proof. Let P be a periodic packing of solid tional over a spectrahedron instead of a for some a ! Rd, where Ln/21- is the spheres of radius r1 /2. This means there is a k polyhedron. That is, we maximize a func- n Laguerre polynomial of degree k with pa- lattice L and points xx1,,f m in R such that tional XX7GH, C over all positive semidef- rameter n/21- (Laguerre polynomials are m inite nn# matrices X that satisfy the lin- a family of orthogonal polynomials). We P = ' ' vx++inBr(/1 2) . ear constraints GHXA, ii= b for im= 1,,f . vL! i = 1^hchoose this form for f , so that its Fourier < a Here GHAB,t= race()BA denotes the transform is The density of P is mB$ vol( n (/r1 2))/ trace inner product. As for linear programs, n vol(R /)L . By Poisson summation we have d semidefinite programs can be solved effi- 2k -r u 2 m fua ()= 1 + / ak u e , (3) ciently by using interior point methods. X fpk = 1 / / fv()+-xxji The usefulness of semidefinite program- ! vLij, = 1 which means that a $ 0 immediately im- ming in solving the above semi-infinite m 1 r $ = fu() e2 iu ()xxji- , plies fua ()$ 0 for all u. Setting r1 = 1 in linear programs stems from the following vol(Rn /)L / V / X uL! * ij, = 1 the above theorem, we see that the optimal two observations: Firstly, Pólya and Szegö and because f()01= and fu()$ 0 for all sphere packing density is upper bounded showed that a polynomial is nonnegative V V u, this is at least mL2 /vol(/Rn ). On the by the maximum of the linear function- on the interval [,1 3) if and only if it can 7 other hand, by the condition fx()# 0 for al afa ()0 over all nonnegative vectors be written as sr12()+-()rs1 ()r , where s1 a ! Rd for which the linear inequalities and s are sum of squares polynomials. x $ r1, we have 2 fra ()# 0 for r > 1 are satisfied. For every Secondly, a sum of squares polynomial of m fixed value ofd this gives a semi-infinite degree 2d can be written as br()< Qb(r), / / fv()+-xxji# mf()0 . vL! ij, = 1 linear program, which is a linear program where br()= (,1 rr,,f d), and where Q Hence, the density of P is at most with finitely many variables and infinitely is a positive semidefinite matrix. (To see many linear constraints. that a polynomial of this form is a sum of fB()02$ vol( n (/r1 )). The density of any packing can be approximated arbitrarily One approach to solving these semi-in- squares polynomial one can use a Chole- < well by the density of a periodic packing, finite programs is to select a finite sample sky factorization QR= R.) Using these so this completes the proof. □ S 3 [,1 3) and only enforce the constraints observations we can introduce two posi- fra ()# 0 for rS! . For each S this yields tive semidefinite matrix variablesQ 1 and We can additionally require either a linear program whose optimal solution a Q2, and replace the infinite set of linear can be computed using a linear program- constraints fr()# 0 for r > 1, by a set of f()01= or r1 = 1, without weakening the a theorem. Moreover, we can restrict to ra- ming solver. Then we verify that fra ()# 0 22d + linear constraints that enforce the dial functions, for if a function f satisfies for all r > 1, or if this is not (almost) true, identity the conditions of the theorem for some r , we run the problem again with a different 1 fr()=-()1 rbr()<

optimal value upper bounds the sphere to magic functions. They parametrized the Viazovska’s breakthrough packing density. For each d this finds the function fa as in (2). Then they required Viazovska made the spectacular discovery optimal function f . that f and f have as many roots and that a magic function indeed exists for a a Va double roots as possible, depending on dimension n = 8. Building on this, Cohn, Producing numerical evidence the degree d. Afterwards they applied Kumar, Miller, Radchenko and Viazovska Now we want to understand what has to Newton’s method to perturb the roots and found a magic function for n = 24. Viazovs- happen when the Cohn–Elkies bound can double roots in order to optimize the val- ka’s construction is based on a couple of be used to prove the optimality of the ue of the bound. In dimension 8 and 24 new ideas, which we want to explain briefly. sphere packing given by an even and uni- they obtained bounds which were too high Each radial Schwartz function f|RRn " modular lattice L. only by factors of 1.000001 and 1.0007071. can be written as a linear combination of Then there exists a magic function This provided the first strong evidence that radial eigenfunctions of the Fourier trans- n n f|RR" and a scalar r1 that satisfy magic functions exist for these two dimen- form in R with eigenvalues +1 and -1. the conditions of Theorem 2, such that sions. Since the magic functions f have to Viazovska wrote the magic function as a the density of the sphere packing equals have infinitely many roots, the degreed linear combination ff=+ab+ f-, where fB()02$ vol( n (/r1 )). As observed before, has to go to infinity. Could this method, f+ is a radial eigenfunction of the Fourier we may assume f()01= , so that r1 is the in the limit, actually give the exact sphere transform with eigenvalue +1 and f- is a shortest nonzero vector length in L. Un- packing upper bounds? radial eigenfunction with eigenvalue -1. der these assumptions on f we can derive The coefficients a and b are determined extra properties that the function f and The Cohn–Kumar paper later on. its Fourier transform must satisfy. Since The next step was taken by Cohn and Ku- She makes the Ansatz that for rr> 1, fx()# 0 and fx()$ 0 for all x $ r , and mar in [2]. They improved the numerical we can write these functions f and f as V 1 + - f()00==() 1, we have equality in the fol- scheme and by using degree d = 803 (with a squared sine function times the Laplace lowing chain of inequalities 3000-digit coefficients) they showed that transform of a (quasi)-modular form. That in dimension 24 there is no sphere pack- is, she proposes that 11##f x = f x . -30 //V ing which is 11+ .65 $ 10 times denser xL!!]]ggxL r 22 than the Leech lattice. The actual aim of fr+ ()=-42sin(/r ) i3 This says that we have to have their paper was to show that the Leech 2 # } (/- 1 zz) ni/22- edr rz z fx()==()x 0 for all xL! \{0}. In fact, we lattice is the unique densest lattice in its # + 0 can apply this argument to any rotation of dimension. For this they used the numeri- L, so that fx()==fx() 0 for all x where and V cal data together with the known fact that x is a nonzero vector length in L. As not- the Leech lattice is a strict local optimum. i3 22 rir2z ed before, we may take f to be radial, and The proof of Cohn and Kumar is a beauti- fr--()=-42sin(/rrz)(# } ),edz then we have (again abusing notation) ful example of the symbiotic relationship 0 between human and machine reasoning in where } is a quasi-modular form and } fr()==fr() 0 + - V . is a modular form. for all nonzero vector lengths r in L. This The sin(/rr222) factor insures (assum- also tells us something about the orders of The Cohn–Miller paper ing the above integrals do not have cusps) the roots. We have f()01= and fr()# 0 For a long time Cohn and Miller were fasci- that the resulting function f (as well as its for rr! [,1 3), so the roots at the vector nated by the properties of these conjectur- Fourier transform) have double roots at all lengths that are strictly larger than r1 must ally existing magic functions. In their paper but the first occurring vector lengths. have even order. We have f()01= and f [4], submitted 15 March 2016 to the arXiv- Viazovska noticed that an analytic ex- V V is nonnegative on [,0 3), so the roots at preprint server, they gave a ‘construction’ tension of f- exists and that it is an eigen- the nonzero vector lengths must have even of the magic functions using function of the Fourier transform having order. of Laguerre polynomials. However, they eigenvalue -1 when the following modu- If f does not have additional roots, then could not prove that this construction in- larity relation holds: in [1] it is shown that there is no other pe- deed worked. With the use of high pre- az + b riodic packing achieving the same density cision numerics they experimented with }}=+()cz dz22- n/ () - b cz + d l - as L. To apply this it is important that E8 their construction. This resulted in im- a b 8 K forall ! SL2 ()Z ,,adodd,,bceven. is the only even unimodular lattice in R proved bounds, optimality of 24 within a eoc d -51 and K24 is the only even unimodular lat- factor of 11+ 0 . Even more importantly, tice in R24 that does not contain vectors they detected some unexpected rational- For the explicit definition of f- we need the theta functions of length 2. ities: For instance using their numerical data they conjectured that for n = 8, the 2 H ()ze=-()1 nir nz The Cohn–Elkies paper magic function f and f have quadratic 01 / V n ! Z In [1] Cohn and Elkies used this insight Taylor coefficients- 27/10 and -32/ , re- and about the potential locations of the roots spectively. For n = 24, the correspond- and double roots to derive a numerical ing coefficients should be- 14347/5460 rin(/+ 12)2z H10 ()ze= / . scheme to find functions that areclose and -205/156. n ! Z 190 NAW 5/17 nr. 3 september 2016 A breakthrough in sphere packing: the search for magic functions David de Laat, Frank Vallentin

Similarly, but technically much more in- E8 K24 volved, the analytic extension of f+ is an ei- genfunction for the eigenvalue +1 when }+ ()EE- E 2 25EE4 -+49 2 EE48 EE2 +-25EE2 2 49EE3 2 is a quasi-modular form. The forbidden Ei- } = 24 6 } = 4 6 464 2 6 2 4 2 + D + D2 senstein series E2 becomes important here. Now it needs to be shown that there exists a linear combination ff=+ab+ f- 55HH12 8 ++HH16 4 2H20 77HH20 8 ++HH24 4 2H28 } 01 10 01 10 01 } 01 10 01 10 01 so that holds, and that - = D - = 2 Yff()00==() 1 D the sign conditions fr()# 0 for rr$ 1 and fu()$ 0 for all u $ 0 are fulfilled. For this, V ri i ri i and more, we refer to the beautiful original fx()=+8640 fx+-() 240r fx() fx()=-113218560 fx+-()- 262080r fx() papers. Here we want to end by taking a look at the magic functions: see Table 2. Table 2 The magic functions.

Interview with Henry Cohn, Abhinav Kumar, Stephen D. Miller and Maryna Viazovska The interview was conducted by Frank A couple more things — in both the our previous numerics in an exact way. Vallentin using the online communication proofs the final inequalities needed for the It’s important to stress that at this point platform Google Hangouts between 20 May functions and Fourier transforms are done we could derive the magic function for 24 and 14 June 2016. with computer assistance. There might be dimensions without using floating point more elegant hand proofs, but so far we calculations, since we had already extrapo- Computer Assistance haven’t found them. And we did quite a lated exact expressions from previous nu- Dear Henry, Abhinav, Steve, and Maryna, lot of messing around with q-expansions et merics. Maybe we will later understand a First of all let me congratulate you to your cetera, which would have been very painful way to derive the 24-dimensional functions breakthrough papers. This issue of the outside of a computer algebra system.” without such information, but at the time it ‘Nieuw Archief voor Wiskunde’ is a spe- Steve: “Though I think there will eventual- was highly convenient to leverage them.” cial issue focussing on computer-assisted ly be a slick proof that can be written by mathematics. In it we have two articles hand, computers were completely essential Maryna, did you also use computer as- about sphere packings: One about the in this story. To me the best example is the sistance when you found the function for formal proof of the Kepler conjecture and appearance of rational numbers that Henry dimension 8? In your paper you mention one about your recent breakthrough on Cohn and I discovered. in passing that one compute the first hun- sphere packings in dimensions 8 and 24. I can only speak for myself, but I was dred terms of Fourier expansions of modu- At the moment a proof of the Kepler con- completely fascinated by the (then pro- lar forms in a few second using PARI/P or jecture without computer-assistance is not posed) existence of these ‘magic’ func- Mathematica. in sight, but your proofs in dimension 8 tions which completely solve sphere Maryna: “Numerical evidence was crucial and 24 require almost no computers. How packing in special dimensions. To satisfy to believe in the existence and uniqueness were computers helpful to you when find- this curiosity, Henry and I began comput- of ‘magic’ functions. I used computer calcu- ing the proofs? ing their features to see if we could learn lations to verify that approximations to the Abhinav: “The Cohn–Elkies paper and lat- more about them. We found — using some magic function computed from linear equa- er the Cohn-Kumar and the Cohn–Miller serendipity with the online ‘inverse sym- tions (similar to the equations considered papers were certainly useful as indicators bolic calculator’ website — that their qua- in the Cohn–Miller paper) converge to the that the solution was out there waiting for dratic Taylor coefficients were rational, function computed as an integral transform the right functions. In our new Cohn–Ku- and furthermore related to Bernoulli num- of a modular form. I used Mathematica and mar–Miller–Radchenko–Viazovska paper at bers. This was a strong hint that modular PARI/GP for this purpose.” least, the numerical data was quite useful forms were connected, though we never because once we had figured out the right understood why until we saw Maryna’s Checking the positivity conditions is the finite dimensional space of modular or paper. We found other rationalities (such only part of the proof which depends on quasi-modular forms, the numerics helped as the derivatives at certain points) us- the use of computers. How do you make us pin down the exact form (up to scaling) ing some theoretical motivation, com- sure that your computer proof is indeed of the function. In particular, we matched bined with good numerical approxima- mathematically rigorous? values and derivatives of f and f at lat- tions. It was a type of ‘moonshine’, with Henry: “In her 8-dimensional proof, Maryna V tice vector lengths to cut down the space a fascinating sequence of numbers and an used interval arithmetic. In the 24-dimen- by imposing linear conditions. Steve has amazing structure we could not otherwise sional case, we used exact rational arithme- a Mathematica code to do some of this access. tic. Either way, it’s not a big obstacle. The but we also used PARI/GP and occasionally Once Maryna’s paper appeared, it took inequalities you need have a little slack, Maple. just a few days to combine her insight with which means you can bound everything David de Laat, Frank Vallentin A breakthrough in sphere packing: the search for magic functions NAW 5/17 nr. 3 september 2016 191

Modular forms For a long time it has been known that

the E8 lattice and the Leech lattice have strong connections to modular forms, sim- ply by their theta series. However, one thing which puzzles us (we mainly work in optimization) is that you solve an opti- mization problem, an infinite-dimensional linear program, with tools from analytic number theory, especially using modular forms. How surprising was it to you that using modular forms was one key to the proof? Henry: “Modular forms are by far the most important class of special functions related to lattices, so in that sense it’s not so sur-

Photo: Archives of the Mathematisches Forschungsinstitut Oberwolfach Forschungsinstitut Mathematisches the of Archives Photo: prising that they come up. Over the years Henry Cohn many people had suggested using them, Abhinav Kumar but it wasn’t clear how. For example, many in any number of different ways, without ture of the summation formulas. For exam- years ago I had tried the Laplace transform needing to do anything too delicate.” ple, Henry and I derived a relevant Fourier of a modular form, but without Maryna’s Steve: “To elaborate: Our positivity check eigenfunction with simple zeros using Vo- sin2-factor. Without that, it seemed impos- involves showing certain power series fq() ronoi summation formulas and derivatives sible to get anything like the right roots, in a parameter q are positive, where q < 1. of modular forms. and therefore the approach was complete- It is not difficult to bound the coefficients However, while these ingredients had ly useless. What I find beautiful about her and deduce this positivity for qc< (where been on the table for a long time, we didn’t proof is how ingeniously it puts every- c is an effective constant), so the problem know how to combine them until Maryna’s thing together (when I know from person- reduces to showing positivity for q in the paper appeared. At that point many of the al experience that thinking ‘I’d better use interval [,c 1]. Numerically one can plot this various pieces of evidence we had sudden- modular forms’ will not just lead you to directly, of course. From such a graph we ly fit together. Quasi-modular forms (which this proof). see fq()> b for some explicit constant b. are derivatives of modular forms) are very Before Maryna’s proof, I could imag- Write fq()=+pq() tq(), where pq() is a crucial to this story.” ine two possibilities. One was that the polynomial consisting of the first sever- right approach would be to solve the LP al terms in the power series and tq() the Collaboration problem in general, getting the excep- tail, with the number of terms chosen so Maryna’s breakthrough paper which solved tional dimensions just as special cases. that tq() is provably less than b/2 for q the sphere packing problem in dimension This approach might not have involved in [,c 1]. We are now reduced to showing 8 was submitted on March 14, 2016 to the modular forms at all. The other was that pq()- b/20> for qc! [,1], and such an arXiv-preprint server. Then it took only one there would be particular special functions inequality can be rigorously established week until you submitted the solution of in those dimensions. I always hoped for using Sturm’s theorem.” the sphere packing problem in dimension something special in 8 and 24 dimensions (e.g., based on the numerical experiments Steve and I worked on), but I was a little worried that maybe the difficulty of writing these functions down indicated that this might be the wrong approach (while solv- ing the problem in general seemed even harder). It was great to see that everything was as beautiful as we had always hoped.” Steve: “The use of Poisson summation in the Cohn–Elkies paper (and an earlier technique of Siegel) had already brought methods of analytic number theory into the subject. It was clear relatively early on that modular forms must be somehow involved in the final answer. This is both because of the appearance of special Ber-

noulli numbers (that prominently arise in Oberwolfach Forschungsinstitut Mathematisches the of Archives Photo: Stephen D. Miller modular forms) as well as the overall struc- Maryna Viazovska 192 NAW 5/17 nr. 3 september 2016 A breakthrough in sphere packing: the search for magic functions David de Laat, Frank Vallentin

24 to the arXiv. Working at five different, search in them. By matching the conjec- ing the finite simple groups, for example.) distant places, how did you collaborate? tured rationalities, we found the even ei- Two dimensions, n = 2, remains open, al- What were the difficulties when going from genfunction on Thursday night and the odd though of course a solution is known by 8 to 24 dimensions? eigenfunction on Friday morning. We used elementary geometry; the LP bounds be- Steve: “That was certainly an exciting and Mathematica and PARI/GP for this. We also have rather differently in two dimensions memorable week. Once we had assembled checked using graphs and q-expansions compared with 8 or 24.” our team, things moved extremely quickly. that the necessary positivity conditions in- Steve: “Yes, dimension two seems very dif- This is mainly because Maryna’s methods deed hold, but did not have a completely ferent because of the delicate arithmetic are so powerful, but it was also import- rigorous proof of this remaining point. nature of the root lengths.” ant that certain pairs of us (Henry–myself, By Friday afternoon I was completely Henry: “I’m sure Maryna’s wonderful ap- Abhinav–Henry, and Danylo–Maryna) were exhausted (and in any event do not work proach to constructing these functions is established collaborators that had already on the Jewish Sabbath). It’s important to just the tip of an iceberg, and I’m opti- worked together well. note that the positivity analysis was a little mistic that humanity will learn more about In addition to phone calls and email, different in 24 dimensions than in 8 be- how LP bounds and related topics work. we used Skype and Dropbox to commu- cause of an extra pole that occurs.” One mystery I find particularly intriguing nicate our ideas. I particularly like draw- is what’s so special about 8 and 24 dimen- ing mathematics on a tablet PC and shar- Going further sions. Maryna’s methods give a beautiful ing the screen on Skype — this allows Now the sphere packing problem has been proof, but I still don’t really know a con- the others to watch as if I’m writing on solved in 1, 2, 3, 8 and 24 dimensions and, ceptual explanation as to what’s different a blackboard. As soon as we saw Mary- coming close to the end of the interview, in, say, 16 dimensions (beyond just the na’s paper on Tuesday morning, we tried it is time to make speculations: Are there fact that the Barnes-Wall lattice isn’t as to make concrete bridges with the numer- candidate dimensions where a solution is nice as E8 or the Leech lattice). On a slight- ical observations Henry and I made in our in sight? Do you think that your method ly different topic, I hope someone solves Cohn-Miller paper. After some reformula- will be useful for this (or for other prob- the four-dimensional sphere packing prob- tion of her modular forms as quotients, lems)? lem, but it will require different techniques. by Wednesday it was then clear what Henry: “It’s hard to say for sure whether Unlike the cases where the LP bounds are properties of the q-expansions would be there might be further sharp cases in high- sharp, these pair correlation inequalities needed to obtain the 24-dimensional func- er dimensions, but it seems unlikely that will not suffice by themselves. But theD 4 tions. We also set up computer programs they would have remained undetected. (At lattice is awfully beautiful, and the world to match potential candidate functions the very least it’s not plausible that they deserves a proof of optimality. The only with the numerical values that we could could have the same widespread occur- question is how... compute separately. rences in mathematics as E8 or the Leech On Thursday we had the right space lattice, since they would presumably have Henry, Abhinav, Steve, and Maryna, thank of modular forms and a program ready to been discovered in the process of classify- you very much for this interview. s

References 1 H. Cohn and N. D. Elkies, New upper bounds 5 J. H. Conway and N. J. A. Sloane, Sphere Kepler conjecture, arXiv:1501.02155 [math. on sphere packings I, Ann. of Math. 157 Packings, Lattices and Groups, Springer, MG], 21 pp. (2003), 689–714. 1988. 8 D. de Laat, F. M. de Oliveira Filho, F. Vallen- 2 H. Cohn and A. Kumar, Optimality and 6 M. Dutour Sikirić, A. Schürmann and F. Val- tin, Upper bounds for packings of spheres of uniqueness of the Leech lattice among lat- lentin, Inhomogeneous extreme forms, An- several radii, Forum Math. Sigma 2 (2014), tices, Ann. of Math. 170 (2009) 1003–1050. nales de l’institut Fourier 62 (2012), 2227– e23, 42 pp. 3 H. Cohn, A. Kumar, S. D. Miller, D. Radchenko 2255. 9 A. Schürmann and F. Vallentin, Local cover- and M. S. Viazovska, The sphere packing 7 T. C. Hales, M. Adams, G. Bauer, D. Tat ing optimality of lattices: Leech lattice ver- problem in dimension 24, arXiv:1603.06518 Dang, J. Harrison, T. Le Hoang, C. Kaliszyk, sus root lattice E8, IMRN 32 (2005), 1937– [math.NT], 12 pp. V. Magron, S. McLaughlin, T. Tat Nguy- 1955. 4 H. Cohn and S. D. Miller, Some properties of en, T. Quang Nguyen, T. Nipkow, S. Obua, 10 M. S.Viazovska, The sphere packing problem optimal functions for sphere packing in di- J. Pleso, J. Rute, A. Solovyev, A. Hoai Thi Ta, in dimension 8, arXiv:1603.04246 [math.NT], mensions 8 and 24, arXiv:1603.04759 [math. T. Nam Tran, D. Thi Trieu, J. Urban, K. Khac 22 pages. MG], 23 pp. Vu and R. Zumkeller, A formal proof of the