The ‘Useless’ Perspective That Transformed Mathematics

Kevin Hartnett:

When representation theory emerged in the late 19th century, many mathematicians questioned its worth. In 1897, the English mathematician William Burnside wrote that he doubted that this unorthodox perspective would yield any new results at all.

“Basically what [Burnside was] saying is that representation theory is useless,” said Geordie Williamson of the University of Sydney in a 2015 lecture.

More than a century since its debut, representation theory has served as a key ingredient in many of the most important discoveries in mathematics. Yet its usefulness is still hard to perceive at first.

“It doesn’t seem immediately clear that this is a reasonable thing to study,” said Emily Norton of the Technical University of Kaiserslautern in Germany.

Representation theory is a way of taking complicated objects and “representing” them with simpler objects. The complicated objects are often collections of mathematical objects — like numbers or symmetries — that stand in a particular structured relationship with each other. These collections are called groups. The simpler objects are arrays of numbers called matrices, the core element of linear algebra. While groups are abstract and often difficult to get a handle on, matrices and linear algebra are elementary.