Background

The sphere packing problem seeks the optimal arrangement of identical spheres in space.
In $\mathbb{R}^8$, this problem was solved by Maryna Viazovska in 2016 (paper), earning her the Fields Medal in 2022.

As part of my undergraduate seminar, I wrote a detailed explanation of this proof which assumes only undergraduate-level knowledge of mathematics (mainly linear algebra and complex analysis). The work was originally written in Hebrew and later translated into English.

• English presentation
• English document
• Hebrew presentation
• Hebrew document

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Seminar on Sphere Packing in Dimension 8

Sphere packing is a geometric problem with a seemingly simple question:
What is the most efficient way to arrange identical spheres in space?

Topics Covered:

• Mathematical Foundations:

  • Definitions of sphere packing, lattices, and density, focusing on the $E_8$ lattice.
  • Modular forms and Eisenstein series.
  • Schwartz functions, Fourier transforms in $\mathbb{R}^n$, and the Poisson summation formula.

• Proof Overview:

  1. Linear programming bounds by Cohn and Elkies (2003).
  2. Constructing a function using a Laplace transform and $\sin(\pi|x|^2/2)$ and manipulate it to find the how the magic function should look like.
  3. Show the construction in Viazovska (2016) of the magic function and proof of its key properties.

Sample Slides:

Building

To build the PDFs, install typst (version 0.12.0), and run the following commands:

typst compile eng/seminar/main.typ eng/seminar.pdf
typst compile eng/seminar_presentation/main.typ eng/seminar_presentation.pdf
typst compile heb/seminar/main.typ heb/seminar.pdf
typst compile heb/seminar_presentation/main.typ heb/seminar_presentation.pdf