That’s Maths: The get-togethers that are essential to solve unsolved problems

Riemann hypothesis is perhaps the most important open problem in maths today

In just three weeks the largest global mathematical get-together will be under way. The opening ceremony of the 2022 International Congress of Mathematicians (ICM) opens on Wednesday, July 6th and continues for nine days. Prior to the ICM, the International Mathematical Union (IMU) will host its 19th general assembly in Helsinki on July 3rd-4th.

The idea of an international congress was first developed by Felix Klein and Georg Cantor, two giants of 19th-century mathematics. The first such meeting was in 1897 and, at the second in 1900, leading German mathematician David Hilbert announced a famous list of 23 key unsolved problems. These had a major influence on mathematical research through the 20th century and several remain unsolved today. At the top of the list is the Riemann hypothesis; perhaps the most important open problem in maths.

The ICM takes place every four years, although this regularity has been disturbed by wars, hot and cold. Once again, war has disrupted, but not derailed, the plans. The 2022 ICM was scheduled to be held in St Petersburg but, on February 26th, following the Russian invasion of Ukraine, the IMU announced the congress would be a fully virtual event, hosted outside Russia. The union acknowledged the dedication of Russian colleagues in preparing for the meeting but concluded an in-person event in Russia was impossible.

Before the congress, the IMU award ceremony will be held on July 5th as a live event in Helsinki, and the winners of the major mathematical prizes will be announced. Most prestigious of these are the Fields Medals, which are awarded to two, three, or four mathematicians who must be under 40 years of age at the international Congress. It is often described as the Nobel Prize of mathematics.

Consistently regarded as the top award in mathematics, the Fields Medal includes a monetary award of about €11,000. Canadian mathematician John Charles Field worked towards designing the award and funding it, but he died before it was established and his plan was overseen by Irish mathematical physicist John L Synge. The first Fields Medals were awarded in 1936.

On the day following the opening ceremony, the Fields medallists will present prizewinners’ lectures. These are open to all who wish to attend and will also be streamed live. There are four slots reserved on the programme. So far, just one woman, Maryam Mirzakhani, has won a Fields Medal, in 2014. Awards were also made that year to Artur Avila, the first South American and to Manjul Bhargava, the first person of Indian origin to receive it.

The virtual ICM is open to all and registration is free of charge on the website mathunion.org. A glance at the session titles indicates the advanced level of the presentations. The topics of several sessions will be familiar: logic, algebra, geometry, number theory, probability and perhaps topology. Others will be less familiar.

Among the many topics that may be found more challenging are algebraic geometry, Lie theory, analysis, dynamics, partial differential equations, combinatorics, numerical analysis, scientific computing, control theory, optimisation, statistics and data analysis, and stochastic and differential modelling. However, the sessions on mathematical education and popularisation, and on the history of mathematics, should include some more accessible presentations.

The general assembly in Helsinki will decide on the venue for ICM 2026. Let us look forward to an in-person event in more peaceful and harmonious circumstances.

Peter Lynch is emeritus professor at the School of Mathematics & Statistics, University College Dublin. He blogs at thatsmaths.com.