The talk will be devoted to the question of sphere packings in higher dimensional spaces. The densest packing in dimension two is given by the hexagonal lattice (that anyone gets almost immediately while packing equal coins on the surface of the table). However, even in dimension 2 it is not at all trivial to prove that this packing is actually the densest one, and in the higher dimensions the question of proving that a packing is actually the densest becomes incredibly hard. Nevertheless, just a few years ago, Maryna Viazovska has proved that the famous E8 (Korkine-Zolotareff) lattice is indeed the densest packing in dimension 8, and just a few weeks later, together with H. Cohn, A. Kumar, S. Miller, D. Radchenko, that the equally famous Leech lattice is the densest packing in dimension 24. In the talk, Dr Kleptsyn will explain the first steps of this path — showing how (at least in principle!) such a result can be obtained; the key element here is an upper bound theorem obtained in early 2000s independently by H. Cohn and N. Elkies and by D. Gorbachev, with a very visual and "geometric" proof.

There will be (almost) no prerequisites; having seen the Fourier series and Fourier transform will help understanding, but the speaker will recall briefly what will be needed.

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