# Topics in Analysis: Sphere Packings, Fourier Analysis and Beyond

Description: In 2016, M. Viazovska announced a major breakthrough in the sphere packing problem, completing its solution in dimension d = 8. Shortly after, Viazovska, with Cohn, Kumar, Miller, and Radchenko completed the solution in dimension d = 24. These two papers almost completed the program launched by Cohn and Elkies in 2001, in which they conjectured the existence of certain extremal functions in Fourier analysis related to the sphere packing problem in dimensions 2, 8 and 24 (the best packing in dimension 2 is known but not the extremal function). These magic functions were indeed found in dimensions 8 and 24 with a beautiful combination of original techniques from Fourier analysis and the theory of quasimodular forms. From the Fourier analysis point of view, the techniques developed by Viazoska and collaborators are remarkable and quite intriguing, leading to a deeper understanding of interpolation/reconstruction formulas with Fourier data, and providing insights in related topics such as uncertainty principles and the theory of quasicrystals.

The aim of this course is to present the basic theory of the Fourier transform, the basic theory of lattices (including some special family of lattices), sphere packings and other associated problems (lattice packings, kissing numbers, codes). We shall discuss the special function approach proposed by Cohn and Elkies to derive upper bounds for the sphere packing density (linear programming bounds) and its connections with quasimodular forms constructions. The intention is to put the audience in position to read (and appreciate) the main papers in the subject-some of them listed in the references below. Our aim is that this course should be self-contained and accessible to a broad audience of researchers and graduate students.

A preliminary list of topics that we intend to discuss in this course includes, but is not limited to:

(1) Elements of Fourier analysis:
(a) Basic Fourier transform theory (L1 and L2 theory);
(b) Lattices (and some geometry of numbers);
(c) Poisson summation over lattices and theta series;
(d) Basic interpolation formulas for bandlimited functions.

(2) Sphere packing general theory:
(a) Proper definitions; existence of the best (uniformly dense) sphere packing;
(b) Lattice packings; kissing numbers; spherical codes;
(c) Special families of lattices.

(3) Papers without linear programming bounds:
(a) Solution of sphere packing in dimension 2 (and maybe 3);
(b) Best lattice packings in dimensions d ≤ 8 and d = 24;
(c) Some non-lattice sphere packing constructions;
(d) Best current asymptotic lower and upper bounds.

(4) Papers with linear programming bounds:
(a) The Cohn and Elkies approach;
(b) The Viazovska papers in dimensions 8 and 24;
(c) Fourier interpolation via quasimodular forms;
(d) Related uncertainty principles.

(5) Laguerre expansions; numerics and algorithms (time permitting).

Referências:
 H. Cohn and N. Elkies, New upper bounds on sphere packings. I. Ann. of Math. (2) 157 (2003), no. 2, 689–714.
 H. Cohn and F. Gon¸calves, An optimal uncertainty principle in twelve dimensions via modular forms, preprint at https://arxiv.org/abs/1712.04438.
 H. Cohn and A. Kumar, Optimality and uniqueness of the Leech lattice among lattices, Ann. of Math. (2) 170 (2009), no. 3, 1003–1050.
 H. Cohn, A. Kumar, S. D. Miller, D. Radchenko and M. Viazovska, The sphere packing problem in dimension 24, Ann. Math. 185 (2017), 1017–1033.
 H. Cohn and Y. Zhao, Sphere packing bounds via spherical codes, Duke Math. J. 163 (2014), no. 10, 1965–2002.
 J.H. Conway and N. J. A. Sloane, Sphere packings, lattices and groups, Third edition. 290. Springer-Verlag, New York, 1999.
 F. Gon¸calves, 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.
 T. Hales, Cannonballs and honeycombs, Notices Amer. Math. Soc. 47 (2000), no. 4, 440–449.
 V. I Levenshtein, Boundaries for packings in n-dimensional Euclidean space, (Russian) Dokl. Akad. Nauk SSSR 245 (1979), no. 6, 1299–1303.
 D. Radchenko and M. Viazovska, Fourier interpolation on the real line, preprint at https://arxiv.org/abs/1701.00265.
 M. Viazovska, The sphere packing problem in dimension 8, Ann. Math. 185 (2017), 991–1015.
 D. Zagier, Elliptic modular forms and their applications. The 1-2-3 of modular forms, 1–103, Universitext, Springer, Berlin, 2008.