The Fields medals are considered the ultimate accolade in mathematics. First awarded in 1936, since 1950 up to four of them have been awarded every four years at the International Congress of Mathematicians, with the latest round due this August in Rio de Janeiro.

The roll call of previous winners, who have to be under the age of 40, includes some of the subject’s most intriguing characters – often unheard of in the wider world.

__Take a look at some of the most intriguing Fields medals past winners:__

**2014 – Maryam Mirzakhani (1977-2017)**

The first – and thus far only – woman to win the Fields Medal, the Iranian Maryam Mirzakhani was honoured for her studies of the geometry of moduli space, a complex geometric and algebraic entity that might be described as a universe in which every point is itself a universe. Already diagnosed with breast cancer at the time of the award, Mirzakhani died last year, aged just 40.

**2010 – Cédric Villani (1973- )**

Villani became something of a celebrity following his 2010 Fields medal win. Once dubbed the Lady Gaga of French mathematicians, that epithet has less to do with his prize-winning work, on the mathematical interpretation of the concept of entropy, than with his distinctive dress sense, often combining ornate velvet cravats with metallic spider brooches.

More approachable than some former winners, and the author of a popular book *Birth of a Theorem: A mathematical adventure*, Villani added another string to his bow in June 2017 when he was elected to the French national assembly as a representative of *La République En Marche!*, the party founded by now-president Emmanuel Macron.

**2006 – Grigori Perelman (1966- ) [declined]**

In 2000, the Clay Mathematics Institute in New Hampshire established a million-dollar prize for anyone who could correctly solve one of seven outstanding problems in mathematics. Eighteen years later, six remain unsolved. The odd one out is the Poincaré conjecture, a 1904 proposal concerning the topology of three-dimensional spheres.

Perelman, a reclusive Russian, finally proved it true in 2002. The significance of his achievement was somewhat overshadowed, however, by his subsequent refusal of the prize – and the Fields medal that followed.

**1998 – Andrew Wiles (1953- ) [silver plaque]**

Wiles was too old to receive a Fields medal when his landmark proof of Fermat’s Last Theorem reached its final form in 1994. At the next round in 1998, he was granted a unique award in recognition of his achievement: a silver plaque.

The theorem states that no three integers a, b, c exist that can satisfy the equation a^{n} + b^{n} = c^{n} where n is greater than 2. (Solutions for n = 2 are easy: 3, 4 and 5, for example, the numbers that make up the sides of a classic “Pythagorean” right-angled triangle.) It had bugged mathematicians ever since 1637, when the French mathematician Pierre de Fermat claimed in a note scribbled in a book margin to have a proof just too long to fit there. If so, we’re missing something: Wiles’s version spanned several hundred pages of cutting-edge 20th century mathematics.

**1990 – Edward Witten (1951- )**

In the words of Michael Atiyah, himself a 1966 Fields medallist, Witten’s “command of mathematics is rivalled by few mathematicians”. Witten is actually a physicist, and his prize was for a mathematical proof of a theorem stemming from Einstein’s general theory of relativity. He is perhaps best known for his subsequent work unifying different flavours of string theory, an attempt to move beyond general relativity to a “theory of everything” that unifies all of nature’s forces. In a less formal award, a poll of physicists attending a cosmology conference the same year saw him dubbed “the world’s smartest physicist”.

**1982 – Alain Connes (1947- )**

**1966 – Alexander Grothendieck (1928-2014)**

Grothendieck’s work in the field of algebraic geometry laid the foundation for much of modern mathematics, including Andrew Wiles’s famed 1994 proof of Fermat’s Last Theorem. It is famously abstract: an obituary written for the scientific journal Nature was nearly rejected when it turned out that almost none of his work could be sufficiently simplified. Some-time New Scientist contributor Richard Elwes sang in a musical tribute in 2016 that it makes less high-powered mathematicians’ “palms go sweaty and knees go weak”.

Grothendieck was an intense man of profound personal conviction. In 1966, he refused to travel to Moscow to collect his Fields medal in protest at the actions of the Soviet regime, and the following year, in response to US involvement in the Vietnam war, went to Hanoi to give mathematics lectures while the bombs fell around him. After retiring from the University of Montpelier in 1988, he retreated to a small village at the foot of the Pyrenees, where he lived in isolation until his death in 2014.

*Original article was published on newscientist.com*