In general, a shape is said to have symmetry when there is some transformation that leaves it the same. To understand a modular form, it helps to first think about more familiar symmetries. And it’s what now makes them crucial to the ongoing development of a “mathematical theory of everything” called the Langlands program. It’s what made them central to more recent work on sphere packing.
It’s what made them key players in the landmark 1994 proof of Fermat’s Last Theorem. The properties that come with those symmetries make modular forms immensely powerful. They are often described as functions that satisfy symmetries so striking and elaborate that they shouldn’t be possible. Every week, new papers extend their reach into number theory, geometry, combinatorics, topology, cryptography and even string theory.
But “there are probably fewer areas of math where they don’t have applications than where they do,” said Don Zagier, a mathematician at the Max Planck Institute for Mathematics in Bonn, Germany. Modular forms are much more complicated and enigmatic functions, and students don’t typically encounter them until graduate school. Part of the joke, of course, is that one of those is not like the others. “Addition, subtraction, multiplication, division and modular forms.” “There are five fundamental operations in mathematics,” the German mathematician Martin Eichler supposedly said.
