Program: In 2022 Maryna Viazovska received the Fields medal. She got it because she proved that a certain highly symmetric sphere packing in R8 is optimal. This packing is associated to the E8 lattice. Two weeks later, in collaboration with others she proved a similar result for R24. This time the packing is associated to the Leech lattice.

    In this school, we lecture on the mathematical ingredients for Viazovska's proof. These are the theory of lattices, sphere packings, Fourier analysis, Laplace transformations, coding theory, modular forms, quasi-modular forms. This material will be covered in the courses by Laura Geatti, Francesco Pappalardi, René Schoof and Peter Stevenhagen.

    If time permits we will go into some details of Viazovska's proof.



    References:
    • H. Cohn, N. Elkies, New upper bounds on sphere packings I, Ann. Math., 157 (2003), 689-714. pdf
    • H. Cohn, A conceptual breakthrough in sphere packing, Notices A.M.S., Providence 2017. pdf
    • H. Cohn, Packing, coding, and ground states. Lecture notes at PCMI 2014. pdf
    • W. Ebeling, Lattices and Codes, Advanced Lectures in Mathematics, Springer 2013. link
    • J. Oesterlé, Densité maximale des empilements de sphères en dimension 8 et 24, Sém. Bourbaki, 2016-17, n. 1133. pdf
    • E. Royer, Quasimodular forms: an introduction. Ann. Math. Blaise Pascal, Volume 19, no 2 (2012), p. 297-306. pdf
    • A. Slipper, Modular magic, Bach. of Arts thesis, Harvard 2018. pdf
    • M. S. Viazovska, The sphere packing problem in dimension 8, Ann. Math., 185 (2017), 991-1015. pdf
    • D. Zagier, Elliptic Modular Forms and Their Applications, In The 1-2-3 of Modular Forms, Universitext, Springer 2008 pdf


    Useful links:
    • J. Oesterlé talk at Seminaire Bourbaki 17/06/2017 video
    • Slides about sphere packings pdf
    • Volume of unit balls and area of unit spheres in Rn png
    • Notes Lattices and Lie algebras pdf


    Exercises:
    • Exercises1 pdf
    • Exercises2 pdf