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💡 Mathematics · Did you know?

Forty-six steps to the almost impossible.

There is an invisible staircase in mathematics. It starts with an exam taken in thousands of Brazilian public schools and ends with seven problems worth one million dollars each. Climb slowly: every step hides a true story.

🏆 Only one person in the world has ever won one of these prizes. And that person turned down the money. Find out who →
1 · Where it all begins 2 · The first thesis 3 · A career 4 · The top
0prizes and honours
0US$ 1 million problems still open
0person has solved one of them
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Before you climb

True or myth?

Take a guess. The answers are scattered along the staircase.

1
Step 1 · the base

Where it all begins

Prizes that exist to find talent where no one was looking.

💡 From the school olympiad to the Fields Medal

Artur Avila started competing in mathematical olympiads while still at school in Rio de Janeiro. In 1995, at 16, he won a gold medal at the International Mathematical Olympiad.

While still a high-school student, and without ever having enrolled at a university, he completed a master's degree at IMPA. He earned his doctorate there too, in 2001. In 2014 he received the Fields Medal, the first Latin American to win it.

🔗 Official source: Brazilian Mathematical Olympiad · IMU · Fields Medals 2014

2
Step 2 · first research

The first thesis

Here, solving hard problems is no longer enough: you have to create mathematics that did not exist before.

💡 The 27-page thesis that became a Nobel

In 1950, John Nash defended a very short doctoral thesis at Princeton, about 27 pages long, called Non-cooperative games. It introduced the idea that now bears his name: the Nash equilibrium.

Forty-four years later, in 1994, those ideas earned him the Nobel Prize in Economics, for the pioneering analysis of equilibria in the theory of non-cooperative games.

🔗 Official source: Nobel Prize · John Nash

3
Step 3 · a lifetime of work

Career recognition

You don't win these for one result, but for decades of them.

💡 The Brazilian who led world mathematics

Jacob Palis, a researcher at IMPA, was Secretary General of the International Mathematical Union from 1991 to 1998 and then its President, from 1999 to 2002. This is the organisation that awards, among other prizes, the Fields Medal.

In 2010 he received the Balzan Prize for Mathematics for his fundamental contributions to the theory of dynamical systems.

🔗 Official source: Balzan Foundation · Jacob Palis

4
Step 4 · very few people per decade

The top of the world

From here on, the names fit on a single sheet of paper.

💡 Why is there no Nobel Prize in Mathematics?

In his will of 1895, Alfred Nobel created five prizes: Physics, Chemistry, Medicine, Literature and Peace. Mathematics was left out, and he never explained why.

The most famous explanation is a legend: Nobel supposedly left mathematics out because of a rivalry with a mathematician. Many people believe it, but no document has ever been found to confirm it. There are other hypotheses, such as the fact that Sweden already had a mathematics prize. The mystery remains, and it is a curious one, because mathematics lies behind almost every field he rewarded.

Another curiosity: the Fields Medal pays about fifteen thousand Canadian dollars, hundreds of times less than the Breakthrough Prize. Even so, it is the one every mathematician dreams of.

🔗 Official source: Nobel Prize · the will

💡 The letter that arrived too late

The Norwegian Niels Henrik Abel died of tuberculosis on 6 April 1829, aged 26, after years of struggling to find a permanent position.

Two days later, on 8 April, a friend wrote from Berlin, unaware of what had happened, to tell him that a permanent post was guaranteed. In 2002 the Norwegian government created the Abel Prize in his honour; it was first awarded in 2003.

🔗 Official source: Abel Prize · Abel's last years

5
Step 5 · prize no. 19

The Millennium Problems

In 2000, the Clay Mathematics Institute chose seven open problems and offered one million dollars for each solution. Six remain locked, and each one leads to its official page. One of them shines. Click it.

🔒 openP versus NPofficial page ↗ 🔒 openHodge Conjectureofficial page ↗ 🔒 openRiemann Hypothesisofficial page ↗ 🔒 openYang–Mills existence and mass gapofficial page ↗ 🔒 openNavier–Stokes existence and smoothnessofficial page ↗ 🔒 openBirch and Swinnerton-Dyer Conjectureofficial page ↗

The Poincaré Conjecture is about the shape of three-dimensional space. It was solved by the Russian mathematician Grigori Perelman, who posted the proof online, in three texts, without submitting it to any journal. It took a few years for the community to confirm that it was correct.

Perelman turned down the million dollars. Before that, he had already declined the Fields Medal and, in 1996, the prize of the European Mathematical Society. He said that the contribution of Richard Hamilton, who created the tools used in the proof, was no smaller than his own, and that taking the glory alone seemed unfair.

🔗 Official source: Clay Mathematics Institute

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Off the main staircase

Prizes for those who dig deep

A parallel circuit: excellence in a single field, or at a single stage of a career.

💡 Ramanujan's lost notebook

In 1913, the Indian mathematician Srinivasa Ramanujan, who had no university degree, sent the mathematician G. H. Hardy in Cambridge a letter full of formulas. Hardy brought him to England the following year. Ramanujan died in 1920, aged 32.

In 1976, the mathematician George Andrews found, in a box of papers kept at the library of Trinity College, Cambridge, pages written by Ramanujan in the last year of his life. They became known as the "lost notebook" and still inspire research today.

🔗 Official source: MacTutor, University of St Andrews

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The learned societies

When peers decide

They don't bring a fortune. They bring something money can't buy: the approval of those who know the subject.

💡 How many women have won the Fields Medal?

Two, in almost ninety years. The Iranian mathematician Maryam Mirzakhani was the first, in 2014, for her study of the geometry of surfaces; she died three years later, aged 40. The Ukrainian mathematician Maryna Viazovska was the second, in 2022, for solving the sphere-packing problem in dimension eight, a question that, in dimension three, stayed open for centuries.

In Mirzakhani's honour, the Breakthrough Prize created an award for women mathematicians who completed their PhDs recently: the Maryam Mirzakhani New Frontiers Prize.

🔗 Official source: IMU · Fields Medals 2014 · IMU · Fields Medals 2022

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Brazil

Brazilian recognitions

Including some that are not only for mathematics, but appear on the CV of anyone doing research in the country.

💡 The Fields Medal that vanished in Rio

In 2018, Rio de Janeiro hosted the International Congress of Mathematicians, the first ever held in Latin America and in the Southern Hemisphere.

At the opening ceremony, the Kurdish mathematician Caucher Birkar received the Fields Medal. Minutes later, the briefcase in which he had put it disappeared. The briefcase was found, but without the medal. The organisers had a spare medal, and a few days later Birkar received another, identical one.

🔗 Official source: IMU · ICM 2018 · IMU · Fields Medals 2018

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Beyond the showcase

Not prizes, but worth gold

Invitations, titles and fellowships that the community reads exactly as it reads a medal.

💡 The most famous lecture in the history of mathematics

In 1900, at the International Congress of Mathematicians in Paris, David Hilbert did something unusual: instead of presenting results, he presented open problems. The published list had twenty-three questions, and they guided much of twentieth-century mathematics.

A hundred years later, the Millennium Problems were inspired by that list. The Riemann Hypothesis appears on both.

🔗 Official source: Clay Mathematics Institute

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There is no Nobel in Mathematics, but…

When mathematicians win other people's prizes

Nobel in Economics · 1994

John Nash

A pure mathematician, made famous by the film A Beautiful Mind. He won the Nobel for game theory, with the Nash equilibrium.

Nobel in Literature · 1950

Bertrand Russell

One of the greatest logicians of the twentieth century and co-author of Principia Mathematica. He received the Nobel for his philosophical and humanitarian writings.

Nobel in Chemistry · 1985

Herbert Hauptman

A mathematician, he created methods to determine the three-dimensional structure of crystals by X-rays, with great impact on chemistry and pharmacology.

Wall Street

James Simons

A geometer, co-author of Chern–Simons theory. He founded Renaissance Technologies and was one of the pioneers of quantitative investing.

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