Universal optimality of and Leech lattices
A set of points in Euclidean -space, energy is a sum of contributions from each pair, which are functions of the distance. What are minimizers?
1. Potential and energy
Fix a potential function . The potential -energy of a finite set is
Motivated by cristallography, we extend the definition to a class of infinite sets.
A point configuration is a non-empty discrete subset of (every closed ball contains only finitely many points of ). Its lower -energy is
If the limit exists, we call it the -energy of .
We say that a configuration has density if
Example. Let be a lattice. The density of is
Furthermore, the -energy of is
Example. Periodic configurations. Fix a lattice . A configuration is -periodic if for every . Such a configuration has density
Our goal is to minimize -energy among configurations of fixed density . We say that minimizes -energy (or is a ground state for ) if its -energy exists and every configuration in with equal density has lower -energy at least .
It sometimes happens that several different potentials admit the same ground states.
2. Universal optimality
How do the ground states depend on ?
2.1. Example: Gaussian core model in
Fix , let
Changing turns out to be equivalent to changing density .
Experiments show that for small densities, the fcc (face centered cubic) lattice wins. For large densities, the bcc (body centered cubic) lattice wins. In 1979, F. Stillinger observed that phase coexistence wins, i.e. fcc in a half space and bcc in the other, with suitable densities, does better than either fcc and bcc.
2.2. Completely monotonic functions
In the sequel, we shall stick to completely monotonic potentials. Say a function is completely monotonic if
- infinitely differentiable,
- derivatives satisfy .
Gaussian kernel, power laws are completely monotonic. Yukawa potential probably not.
Definition 1 (Cohn, Kumar 2007) Say a configuration in with density is universally optimal if it minimizes -energy for every potential ofthe form , where is completely monotonic.
Theorem 2 (Bernstein) Every completely monotonic function can be written as an integral
for some positive measure on .
Corollary 3 is universally optimal minimizes all energies defined by Gaussian core models.
Theorem 4 (Cohn, Kumar 2007) is universally optimal.
They conjectured that the lattice in , the lattice in and the Leeach lattice
The main result of the present lectures is
Theorem 5 (Cohn, Kumar, Miller, Radchenko, Viazovska 2019) and the Leech lattice are universally optimal.
is still open.
The sphere packing theorem follows by a limiting argument.
3. Strategy of the proof
The key are linear programming bounds. These amount to relax the problem: replace it by a series of convex problems on functions on . It becomes infinite dimensional. Fourier interpolation enters the scene. This reduces the problem to the positivity of a function of 2 variables. The last step is handled numerically.
3.1. Linear programming bounds
A Schwarz function on is
- infinitely differentiable,
- each derivative decays faster than any inverse polynomial.
Our choice of normalization of Fourier transform is
The following linear programming formulation is due to Cohn and Kumar 2007.
Proposition 6 Let be a nonnegative potential. Let be a Schwarz function such that
Then every configuration of with density has lower -energy at least .
This reduces the problem to finding a clever radial Schwartz function . Note that the assumptions on are convex.
3.2. Proof of Proposition 6
I explain the case of periodic configurations. Let be a -periodic configuration. Then
The general case (nonperiodic) was proved by Cohn and Courcy-Ireland. Every configuration can be viewed as a limit of periodic ones.
3.3. When can we hope for a sharp bound?
Did we lose much? Suppose that a lattice minimizes -energy for some nonnegative potential . Suppose that this can be proven by linear programming using a Schwartz function . There is no loss in the above inequalities
- on .
- on .
We see that firqt derivatives must satisfy the above requirements as well. Can we recnstruct a Sxhwartz function from these conditions?
3.4. Fourier interpolation
Theorem 7 Let or . Then any radial Schwartz function is uniquely determined by the values of
for integer .
More precisely, there exists an interpolation basis , , , , consisting of radial Schwartz functions, such that for every radial Schwartz function and every ,
where the series converges absolutely. Here are properties the interpolating basis.
Denote by the and Leech lattices respectively. Note that the shortest vector of , or , is .
3.5. Construction of the optimal auxiliary functions
The only possible auxiliary function that could prove a sharp bound for , (among radial Schwartz functions) would be the following one,
Recall that, in addition to equality, one must prove inequalities
- on .
- on .
Therefore it suffices to check that
- on .
- on .
Here, is a Gaussian.
3.6. Generating functions
Substitute . Then the inequalities
for all and and
for all and , imply the universal optimality of .
Apply the interpolation formula (IF) above to . It is equivalent to
We secretly know that
Note that the condition implies a control on the growth:
Indeed, using the functional equations, we get and
4. Relating the Fourier interpolation formula with functional equations
4.1. Summary of previous episodes
Recall our goal:
Theorem 8 (Cohn, Kumar, Miller, Radchenko, Viazovska 2019) Let or . There exists an interpolation basis , , , for integer , consisting of radial Schwartz functions, such that for every radial Schwartz function and every ,
where the series converges absolutely.
Theorem 9 (Cohn, Kumar, Miller, Radchenko, Viazovska 2019) Consider the maps
These maps are isomorphisms and inverses of each other.
Again, we form the generating series
The interpolation formula implies the following functional equations.
4.2. Reversed procedure: functional equations imply interpolation formula
Conversely, we shall now check that the functional equations imply the interpolation formula.
Notations. For a radial Schwartz function , expressed as , let .
Lemma 10 The complex Gaussians span a dense subspace in . For any , the same is true if we consider only complex Gaussians with .
Therefore the interpolation formula needs be proven only for complex Gaussians.
Here comes the converse statement.
Theorem 11 Suppose there exist functions , such that
- and are holomorphic in upper half plane.
- and are radial in .
- For all ,
In the special case , we make a stronger assumption
for , with .
- and satisfy the three functional equations above.
Then and have expansions of the form above (with ), for some radial Schwartz functions . Moreover, for every radial Schwartz function , the interpolation formula
holds. Finally, for every , the radial seminorms , …, grow at most polynomially in . The n=1 functions ,… vanish if and only if and are in the strip with fixed.
4.3. Proof of Theorem 11
Step 1. As functions of , is holomorphic, invariant under translation by 1 and goes to as tends to . Therefore it can be viewed as a function (of ) on the disk, hence it admits an expansion in powers of , with coefficients which we denote by .
Same argument with provides the coefficients. The same for .
Step 2. The functions ,… are radial Schwartz functions.
Express as a Fourier coefficient,
and idem for . Part (3)of the assumptions, we see that the radial seminorms of … are finite. To show that they grow polynomially, we take in the integral. This moderate growth implies that the interpolation series converges absolutely for every radial Schwartz function . This defines a continuous linear functional on .
Step 3. Proof of the interpolation formula.
Fix and consider the linear functional
By density, we need merely prove that vanishes on complex Gaussians. This is equivalent to the first functional equation for and . This concludes the proof of Theorem 11.
Remark. Suppose that and are solutions of the homogeneous equations (i.e. removing right hand sides)
The proof of Theorem 11 shows that for any radial Schwartz function ,
This shows that the orthogonal of the image of is the finite dimensional space of solutions of the set of homogeneous functional equations.
5. Construction of solutions of functional equations: preliminary steps
How to construct and ?
Remember that the group acts on the upper half plane .
Definition 12 Let be a function, let be an even integer. Fix an element . The slash operator is defined by
where is the denominator of .
It has the following property: .
Notation. Let denote the group algebra (with the right action of the group). The slash operator extends to by linearity.
Definition 13 A function on has moderate growth if there exist such that
The space of holomorphic functions with moderate growth is denoted by .
is invariant under the slash operator.
Notation. The involution and the translation by are the usual generators for , with relators and .
The functional equation for and can be written
We use the last equation to express in terms of and reduce to a set of 2 equations with only one unknown function,
We want to solve this equation in , and eventually find a unique solution.
5.2. The ideal associated to functional equations
In the group algebra , consider the right ideal
It has the following special properties.
- has complex codimension in .
- is the free -module generated by and .
- The right action of on the -dimensional complex vectorspace has “polynomial growth”.
The last statement means that in a basis of , the absolute values of the matrix coefficients of an element are bounded above by powers of the matrix coefficients of . It follows from the following description of the representation . Let denote the restriction to of the -dimensional representation of . Let denote the -dimensional representation of such that the kernel of is the congruence subgroup (its image has elements, it is isomorphic to the dihedral group ). Let denote the cocycle (with respect to ) such that and . This defines an affine action, therefore a morphism to . Up to conjugacy, the -dimensional representation is the direct sum of this morphism and .
5.3. Next episode
Tomorrow, we shall solve the homogeneous functional equations (using some classical theory of modular forms), then solve the nonhomogeneous functional equations in the form
where is meromorphic on , with special requirements on residues at poles (which are the fixed points of ). It satisfies
(with respect to the variable) for all .
6. Solving the functional equations
Definition 14 Let be a right-ideal of the group ring . For an even integer , denote by
We shall describe in terms of modular forms.
6.1. Brief overview of classical modular forms
Definition 15 Let be a discrete finite covolume subgroup of . A (holomorphic) modular form of weight is a function such that for all . They form a finite dimensional space .
We also define the infinite dimensional space of weakly holomorphic modular forms by admitting poles near cusps.
We are interested in the cases and , the principal congruence subgroup at level .
6.2. Modular forms for
Eisenstein series are defined for by
Its Fourier expansion is
Then, as an algebra, the direct sum of all is freely generated by and .
Ramanujan’s cusp form of weight is
does not vanish on , its inverse is a weakly holomorphic modular form of weight . vanishes at all cusps. Another famous weakly modular form is the -modular invariant . It is a Hauptmodul for the modular surface .
is generated by and .
One can also define Eisenstein series of weight , but not by the series, by the following formula instead,
It satisfies and
It is called a quasimodular form, and more generally, one calls quasimodular form all elements of the algebra generated by , and .
6.3. Modular forms for
We start with Jacobi theta functions
These are modular forms of weight , we shall merely need their powers
which are honest modular forms. They satisfy the Jacobi identity,
Then the modular ring is generated by and , the weakly modular ring is generated by and .
The modular -function is . It takes its values in .
It is a Hauptmodul of the modular curve , i.e. it generates its function field over .
The fundamental domain of is the union of two adjacent fundamental domains of . The function descends to an isomorphism of with , extending to a compactification by , and .
One denotes by and .
6.4. Role of ideal
We introduce two other ideals
They have codimension in , their sum is and their intersection is . An alternate description is
It follows that
We shall treat both pieces in different manners, the first in terms of quasimodular forms, the second in terms of logarithmic derivatives of the -function.
Proposition 16 Let be an even integer. Then
It follows that
Proposition 17 Let be an even integer. Then
and , .
It follows that
Comment. It is fortunate that solutions can be expressed in modular terms. It is folklore that the symmetric square representation of is related to modular forms. It is not so folklore for the affine representation. However, it is not that dramatic: we knew a priori that the annihilators were finite dimensional. In case of need, we could have used numerical calculations to describe them.
6.5. Solving the inhomogeneous equation
We shall use again the fundamental domain for ,
Here are premiminary technical points.
Proposition 18 Let be continuous functions on .
- Analytic continuation. Assume are holomorphic. Let be an open neighborhood of . Let be a holomorphic function which satisfies the following transformation laws:
whenever both sides are defined. Then extends to a holomorphic function on which satisfies the same equations (a) and (b).
- Propagation of moderate growth bounds. Suppose that is a continuous function on which satisfies equations (a) and (b). Assume that has moderate growth on , and that and have moderate growth. Then has moderate growth on .
Now I describe our ansatz. We look for solutions in the form
Theorem 19 For dimensions , there exist unique meromorphic functions on satisfying the following properties:
- For a fixed , the poles of in are all simple and contained in the -orbit of .
- The functional equations hold,
- For and ,
where is the linear map defined by its values on the basis as follows,
- The functions
are in the class both as functions of and . Moreover
as tends to infinity.
The residue constraint is found by plugging in the answatz, switching contours and fitting residues.
6.6. Idea of the proof of Theorem 19
The proof consists in providing explicit formulae for these functions. Assume that a solution exists.
The group ring comes equipped with an involution such that for group elements. The slash operation, when performed on the variable, relates to the usual slash operation.
Lemma 20 Given matrices ,
We introduce one more ideal,
Fix . For all ,
Therefore is holomorphic in the variable. Growth conditions imply that belongs to , hence to as a function of . We also knowthat this function vanishes at cusps. This implies that for all .
Now we can describe the spaces . We use the fact that
for suitable , this gives us an explicit formula for in terms of Eisenstein series.
For and , (this is required for the integral to converge), remember we set
Lemma 21 The function extends to a holomorphic function on an open neighborhood of in , and satisfies