Lectures and study topics

Topic 1: Sign uncertainty, interpolation and quasicrystals - Felipe Gonçalves

Content of the lectures

First, we will give a brief introduction on the interpolation theorems for the band-limited case and discuss interesting applications. Secondly, we will talk about the sign uncertainty principle of Bourgain, Clozel and Kahane, its solution in dimension 12 and some generalizations to other spaces and weights. Thirdly, we will discuss the construction and stability of the Fourier interpolation formulas of Radchenko and Viazovska. Finally, we will discuss some recent advances in the characterization and construction of crystalline measures.

Topics for student groups

The following topics of study for the working groups are suggested and will be coordinated by Felipe Gonçalves.

Topic 1.1:  The one dimensional sign uncertainty principle

The one dimensional +1 sign uncertainty principle still is unsolved, in contrast, the  ́1 sign uncertainty has infinitely many solutions in dimension 1, but some very simple ones. The goal in this project is to make advances in the solution of the +1 uncertainty principle, and perhaps find a solution. Numerical data via the discrete model indicates there is an extremizer that vanishes identically at infinitely many intervals and the dual problem has the same characteristics.

Topic 1.2:  Asymptotics Upper Bounds for Packings

The Kabatiansky and Levenshtein upper bounds for sphere packings state that no sphere packing in dimension n has the logarithm of its density larger than -0.599n and this bound has not been improved since 1978. Recent numerical experiments have shown it can be improved via linear programming, but only slightly, to -0.604n. The goal of this project is to understand why their construction works so well in light of  the recent developments in the area and possibly try to improve it.

References

[1] J. Bourgain, L. Clozel, and J.-P. Kahane. Principe d’Heisenberg et fonctions positives. Ann. Inst. Fourier (Grenoble) 60 (2010), no. 4, 1215–1232.

[2] H. Cohn and F. Gonçalves. An optimal uncertainty principle in twelve dimensions via modular forms. Invent. Math. 217 (2019), no. 3, 799–831.

[3] H. Cohn and Y. Zhao. Sphere packing bounds via spherical codes. Duke Math. J. 163 (2014), no. 10, 1965–2002.

[4] F. Gonçalves, D. Oliveira e Silva, and J. P. G. Ramos. On regularity and mass concentration phenomena for the sign uncertainty principle. J. Geom. Anal. 31 (2021), no. 6, 6080-6101.  

[5] F. Gonçalves, D. Oliveira e Silva and J. P. G. Ramos. New sign uncertainty principles, available at arXiv:2003.10771.

[6] F. Gonçalves, D. Oliveira e Silva, and S. Steinerberger. Hermite polynomials, linear flows on the torus, and an uncertainty principle for roots. J. Math. Anal. Appl. 451 (2017), no. 2, 678–711.

[7] G. A. Kabatiansky and V. I. Levenshtein. Bounds for packings on a sphere and in space (in Russian). Problemy Peredachi Informacii 14 (1978), 3–25; English translation in Probl. Inf. Transm. 14 (1978), 1–17.

[8] N. Lev, N. and A. Olevskii. Quasicrystals and Poisson’s summation formula. Invent. Math. 200 (2015), no. 2, 585-606.

[9] J. P. G. Ramos and M. Sousa. Perturbed interpolation formulae and applications. To appear in Anal. & PDE. 65 pages.

[10] J.  P .G. Ramos and M. Stoller. Perturbed Fourier uniqueness and interpolation results in higher dimensions. J. Funct. Anal. 282 (2022), no. 12, 34 pp.

Topic 2:  Optimization methods for sphere packing problems - Philippe Moustrou

Content of the lectures

The main goal of this course is to explain how convex optimization techniques can be used to provide upper bounds for sphere packing problems.

After an introduction about linear and semi-definite programming (LP and SDP), we will see how to derive LP and SDP bounds for several sphere packing problems, both on the sphere and in the Euclidean space, by writing these problems as independence number problems for infinite graphs. This will cover the original LP bound for spherical codes due to Delsarte, Goethals and Seidel, the Cohn-Elkies bound for sphere packings in the Euclidean space, and the three-point bound by Bachoc and Vallentin.

Then, we will discuss how to compute numerical bounds in practice from these infinite-dimensional linear or semi-definite programs. Finally, when these bound are sharp, they might lead to proofs for optimality and uniqueness of specific configurations. We will finally study what are the challenges to obtain such a proof starting from a bound that computations suggest to be sharp.

Topics for student groups

The following topics of study for the working groups are suggested and will be coordinated by Philippe Moustrou.

Topic 2.1: Universal optimality of point configurations on spheres

When dealing with spherical codes, one is looking for optimal point configurations when the constraints are only given by the closest pairs of points. One can consider more general interactions: given a potential function depending on the distance between points and a number of particles, how to arrange the particles on a sphere in order to minimize the global potential energy of the configuration? In this topic, after making a connection between this concept and sphere packings, we will study the notion of universally optimal configurations, minimizing a large class of potential functions. In this context, we can naturally adapt the standard bounds for packings described in the course. However, in order to prove universal optimality of a configuration, one needs to solve exactly infinitely optimization problems. The main goal of this project consists in understanding how to prove such results, using general interpolation formulas and non-negativity certificates for infinitely many potential functions.

Topic 2.2: Hierarchies of semidefinite programming bounds in discrete geometry

This projects gives a better understanding on the connection between 2pt and 3pt bounds described in the course, by seeing them as the first steps of a hierarchy of SDP bounds. After understanding the main ideas of the Lasserre hierarchy for the independence number of finite graphs, we will see how it can be generalized to infinite graphs, and applied to different packing problems in discrete geometry. If the main focus on the topic is sphere packing, if times permits, one can also look at other applications of such hierarchies, for example in the contexts of error-correcting codes or equiangular lines.

References

[1] C. Bachoc, F. Vallentin. New upper bounds for kissing numbers from semidefinite programming.  J. Amer. Math. Soc. 21 (2008), no. 3, 909–924. 

[2] E. Bannai and Neil J. A. Sloane. Uniqueness of certain spherical  codes. Canad. J. Math., 33 (1981), no. 2, 437–449.

[3] H. Cohn and N. Elkies. New upper bounds on sphere packings I. Ann. of Math. (2) 157 (2003), no. 2, 689–714. 

[4] H. Cohn and A. Kumar. Universally optimal distribution of points on spheres. J. Amer. Math. Soc. 20 (2007), no. 1, 99–148.

[5] H. Cohn, A. Kumar, S. D. Miller, D. Radchenko, and M. Viazovska. The sphere packing problem in dimension 24. Ann. of Math. (2) 185 (2017), no. 3, 1017–1033.

[6] H. Cohn and J. Woo. Three-point bounds for energy minimization. J. Amer. Math. Soc. 25 (2012), no. 4, 929–958.

[7] D. de Laat, W. de Muinck Keizer, and F. C. Machado. The Lasserre hierarchy for equiangular lines with a fixed angle, available at arXiv:2211.16471.

[8] D. de Laat and F. Vallentin. A semidefinite programming hierarchy for packing problems in discrete geometry. Math. Program. 151 (2015), no. 2, Ser. B, 529–553. 

[9] P. Delsarte, J.M. Goethals, and J. J. Seidel. Spherical codes and designs. Geometriae Dedicata 6  (1977), no. 3, 363–388.

[10] G. A. Kabatiansky and V. I. Levenshtein. Bounds for packings on the sphere and in space. Problemy Peredači Informacii 14 (1978), no. 1, 3–25. 

[11] J.B. Lasserre. An explicit equivalent positive semidefinite program for nonlinear 0-1 programs. SIAM J. Optim. 12 (2002), no.3, 756–769. 

[12] M. Laurent. A comparison of the Sherali-Adams, Lovász-Schrijver, and Lasserre relaxations for 0-1 programming. Math. Oper. Res. 28 (2003), no. 3, 470–496. 

[13] M. S. Viazovska. The sphere packing problem in dimension 8. Ann. of Math. (2) 185 (2017), no. 3, 991–1015.

Topic 3: Modular forms and their applications to sphere packings - Danylo Radchenko

Content of the lectures

In this mini-course I will give an introduction to the theory of modular forms aimed at applications to the sphere packing problem and related questions. 

In addition to basic notions about modular forms, I will cover classical theory of theta functions of lattices, give a quick overview of quasimodular forms, and explain the construction of magic functions for the sphere packing problem that was given by Viazovska [2] for d=8 and by Cohn, Kumar, Miller, Radchenko and Viazovska [3] for d=24.

I will also talk about modular integrals and their relation to Fourier interpolation formulas that have been used in [4] to prove universal optimality of the E8 and the Leech lattice.

Topics for student groups

The following topics of study for the working groups are suggested and will be coordinated by Danylo Radchenko.

Topic 3.1: Properties of the magic functions

The so-called magic functions are functions that give tight L^p bounds for the sphere packing problem in dimensions 8 and 24. Their existence was conjectured by Cohn and Elkies in [6], but before their construction was found, Cohn and Miller [7] have observed several curious properties that they seemed to have, based on numerical data. The goal of this project is to understand as much as possible about the magic functions (analytic continuation, singularities, growth, etc.) from their explicit construction using modular forms given in [2], [3], and to verify some of the conjectures from [7]. Another, more ambitious goal would be to find a conceptually more satisfactory proof of the key inequalities for the magic functions, at least for d=8.

Topic 3.2: Extremal lattices

The best sphere packings in dimensions 8 and 24 are given by extremal even unimodular lattices, and the same is true for the current best known packing in dimension 48. A d-dimensional even unimodular lattice is called extremal if it doesn't have any vectors of squared norm 2,4,...,2m-2, where m is the dimension of the space of modular forms of weight d/2 and level 1. Constructing extremal lattices in moderately big dimensions is a difficult problem, but Mallows, Odlyzko, and Sloane [5] have proved that extremal unimodular lattices don't exist in all sufficiently large dimensions, and the goal of this project would be to understand the proof of their result. Another goal is to understand the proof of Siegel's result that an even unimodular lattice always has a vector of squared norm 2,4,...,2m-2, or 2m, that relies on similar ideas, but it is technically easier.

References

[1] H. Cohn and N. Elkies. New upper bounds on sphere packings. I, Ann. of Math. (2) 157 (2003), no. 2, 689–714. 

[2] H. Cohn, A. Kumar, S. D. Miller, D. Radchenko, and M. Viazovska. The sphere packing problem in dimension 24. Ann. of Math. (2) 185 (2017), no. 3, 1017–1033. 

[3] H. Cohn, A. Kumar, S. D. Miller, D. Radchenko, and M. Viazovska. Universal optimality of the E8 and Leech lattices and interpolation formulas. Ann. of Math. (2) 196 (2022), no. 3, 983–1082.

[4] H. Cohn and S. D. Miller. Some properties of optimal functions for sphere packing in dimensions 8 and 24, available at arXiv: 1603.04759.

[5] C. L. Mallows, A. M. Odlyzko, and N.J.A. Sloane. Upper bounds for modular forms, lattices and codes. J. Algebra 36 (1975), no. 1, 68–76. 

[6] M. S. Viazovska. The sphere packing problem in dimension 8.  Ann. of Math. (2) 185 (2017), no. 3, 991–1015. 

[7] D. Zagier. Elliptic modular forms and their applications. The 1-2-3 of modular forms, 1–103, Universitext, Springer, Berlin, 2008.