### The Abel Prize – The Nobel Prize for Mathematics

There is no Nobel Prize for mathematics, but there is a close equivalent: The prestigious Abel Medal is awarded every year for outstanding work in mathematics [TM086, or search for “thatsmaths” at irishtimes.com]. This years winner, or winners, will be announced soon.

When Alfred Nobel’s will appeared, the absence of any provision for a prize in mathematics gave rise to rumours of discord between Nobel and Gösta Mittag-Leffler, the leading Swedish mathematician of the day. They are without foundation, the truth being that Nobel had little interest in the subject, and probably didn’t appreciate the practical benefits of advanced mathematics.

The lack of a Nobel Prize for mathematics eventually led to the establishment, by the Government of Norway, of a major prize, to be awarded annually for outstanding work in the field. The winner of the 2016 Abel Prize will be announced on 15 March. The prize, to be presented by King Harald V of Norway, is the most prestigious award in mathematics. The prize medal is accompanied by a sum of six million Norwegian crowns, comparable to the value of the Nobel awards.

Niels Henrik Abel

But who was Abel? Born in 1802, Niels Henrik Abel made profound contributions to mathematics in an all-too-brief life. Perhaps his most important achievement was a proof that there is no algebraic formula for solving quintic equations. A linear equation, like 2x – 6 = 0, has one root, that is, one value of x that makes it hold true. This root is easily found: add 6 to both sides and divide by two to get x = 3. A quadratic equation, like x² – 4 x – 5 = 0 contains the square of x, and has two roots. We can find them by using the formula that we learned in school.

Italian renaissance mathematicians found more complicated formulas for cubic equations which involve the third power of x and quartics, involving the fourth power. But for centuries, mathematicians struggled to find such a formula for quintic equations, which involve the fifth power of x. It was Abel who first showed that such a formula is impossible.

Abel grew up in difficult times of widespread famine in Norway. He was far from the mathematical centre of action, but he was fortunate to have an inspiring teacher, Bernt Holmboe, who was familiar with current developments in European mathematics. Abel soon surpassed his teacher, producing results of startling originality.

Abel travelled to Germany and France seeking recognition for his work. In Germany he had some success, meeting August Crelle, who published his work in a new journal. But he was less fortunate in France, where he sent a manuscript to the renowned Augustin-Louis Cauchy, who lost it. Worse still, whilst in Paris, Abel contracted tuberculosis. This led to his untimely death two years later when he was just 26 years old.

A sample form Abel’s notebook

Abel did not live to see his brilliant work receive recognition. But with the posthumous publication of his collected works, the great significance of his contribution to mathematics became clear. He is remembered today in the adjective “abelian, which is applied to several mathematical objects: groups, categories and varieties.

The 2016 Abel Prize will be the fourteenth award, the first having been presented in 2003. I will write at a later stage about the winner and – if I can understand it – about what he or she has done.

Official Abel Prize Website:  http://www.abelprize.no/