On December 21 and 22 at a function held in Kumbakonam, in Tamil Nadu, Maryna Viazovska, Ukrainian mathematician who is currently based at Swiss Federal Institute of Technology, Lausanne, was awarded the SASTRA Ramanujan Award. She was given the award for having solved the problem of what is the densest possible way of stacking of spheres in eight and 24 dimensions. The problem is centuries old and Prof. Viazovska’s proof involved ingenious use of the so-called modular forms – one of Srinivasa Ramanujan’s favourite topics. The prize is given annually to mathematicians under 32 who have made a remarkable contribution to mathematics in areas related to Ramanujan’s work and ideas. Other mathematicians who have been given this award have gone on to win awards like the Fields Medal and the Cole prize.
The chairperson of the SASTRA Ramanujan Award Committee and professor of mathematics at University of Florida, U.S., Krishnaswamy Alladi says, “There were four contenders for the award, but Maryna Viazovska was chosen because the problem itself is such a longstanding one and she has come up with such an ingenious solution.”
The densest packing of spheres in space was known only in dimensions 0, 1, 2 and 3 until Viazovska proved it for dimensions eight and 24.
If we had a set of oranges and wanted to stack them on a table most efficiently, we would automatically stack them first on the table in an alternating arrangement so that every orange has six nearest neighbours. This is the problem in two-dimensions, the plane of the table. If we wish to make a three-dimensional stacking, we would place the second layer on top of the tiny spaces we see between three neighbouring oranges. Building layer by layer in this manner, we would be able to solve the packing problem in three dimensions.
National Mathematics Day: (Left) Maryna Viazovska receiving the SASTRA Ramanujan Award 2017 from Badrinarayanan Parthasarathy, Vice-President, TATA Communications, (right) efficient stacking of fruit .
However, while it is easy to intuitively judge this, we cannot prove that this is the best way and no other way of stacking oranges can match this. The seventeenth century mathematician Johannes Kepler, famous for having given the description of planetary orbits, was intrigued by this problem in three dimensions. But the proof of the packing problem in three dimensions had to wait until 1998 when Thomas Hales, now at Pittsburgh, proved it mathematically.
The packing problem’s proof for dimensions higher than three is further complicated because once the dimension increases the interstices between the spheres become so large that you can put additional spheres in it.
Leech lattice
Maryna Viazovska has shown that if you place the spheres such that their centres lie on the points of what is called an E8 lattice, then that is the densest stacking in eight dimensions. In 2001, Henry Cohn of Microsoft Research, Cambridge, and Noam Elkies, Harvard University, showed that the E8 lattice came close to being the densest packing in eight dimensions. They “conjectured,” or guessed, the existence of some magic functions which can resolve the problem. In fact, they also talked about dimension 24 in the same vein.
In what has been described as an audacious attempt, Prof. Viazovska used the so-called modular forms very creatively to find these functions, first in 8 dimensions and then, in collaboration with others, in 24 dimensions. If there is the E8 lattice in 8 dimensions, there is the so-called Leech lattice in 24 dimensions, and these were shown by her to have the densest packings in eight and 24 dimensions, respectively.
Prof. Viazovska’s papers have been published in the Annals of Mathematics.
