Brilliant mathematical research honoured

Every four years, at the International Congress of Mathematicians, the Fields Medal is awarded to two, three, or four young mathematicians. To be eligible, the awardees must be under 40 years of age. For the chosen few, who came from England, France, Korea and Ukraine, the award, often described as the Nobel Prize of Mathematics, is the crowning achievement of their careers.

The congress, which ran from July 6th to 14th, was originally to take place in St Petersburg. When events made that impossible, the action shifted to Helsinki and the conference presentations were moved online. The International Mathematical Union generously allowed participants to register at no cost.

Prime gaps and prime clusters

James Maynard, professor of mathematics at Oxford University, was honoured for his “spectacular contributions to analytic number theory”. He has made ground-breaking contributions to our understanding of the distribution of prime numbers.

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Recall that a prime number is one that cannot be constructed by multiplying smaller numbers together; the primes are the atoms of the number system. Primes get relatively scarce as we move to greater magnitudes, but the distribution of these numbers is extraordinary. As observed by Don Zagier, they “grow like weeds among the natural numbers in an apparently haphazard way”, and yet they exhibit stunning regularity. Despite efforts over millennia to fathom their behaviour, we are still far from a complete understanding.

In late February, just weeks after she learned she was to be honoured, Russian tanks and planes began pounding Kyiv, the home city of Maryna Viazovskai, now professor at the Ecole Polytechnique in Lausanne, She is only the second woman in the medal’s 86-year history to receive a Fields Medal; Maryam Mirzakhani was the first, in 2014. Viazovskai used modern mathematical methods to explore a centuries-old question: how to pack spheres together in the densest possible way.

Johannes Kepler, the great astronomer, was among the earliest scholars to tackle the packing problem. We are all familiar with the honeycomb pattern, the densest way to pack circles in the plane, and the pyramidal stacks of oranges, the optimal solution in three dimensions. As the culmination of 13 years of study, Viazovskai found an explicit formula describing the best packing solution in high dimensions. Its bound matches the E8 lattice perfectly in eight dimensions, and also the Leech lattice in 24 dimensions, proving them to be the densest packings.

Maths-physics interface

Hugo Duminil-Copin, based at the University of Geneva, was awarded the Fields Medal for solving problems of phase transitions. His work straddles the interface between mathematics and physics. Physicists have grappled with phase transitions, seeking a full understanding.

A glass of water placed in a freezer cools smoothly but then undergoes an abrupt transition from liquid to ice. In statistical physics, we seek to explain the behaviour of complex systems like this by analysing interactions between tiny components. Duminil-Copin’s work was inspired by analogies with fluid percolating through a porous medium, or water straining through coffee grounds and the interactions of tiny dipoles in a magnetic material.

June Huh grew up in Korea. As a youth, he dreamed of becoming a poet. Now a professor of maths at Princeton University, he has proved several conjectures in combinatorial theory, including one called the Dowling-Wilson Conjecture. Huh’s research focuses on geometry, topology, and combinatorics. The citation noted, “Huh, with his collaborators, has transformed the field of geometric combinatorics”. We may surmise that poetry and maths are not so far apart.

• Peter Lynch is emeritus professor at UCD School of Mathematics & Statistics — he blogs at thatsmaths.com