Harmonic analysis

Amid the chaos of revolutionary France, one man’s mathematical obsession gave way to a calculation that now underpins much of mathematics and physics. The calculation, called the Fourier transform, decomposes any function into its parts.

After finding the homeschooling life confining, the teen petitioned her way into a graduate class at Berkeley, where she ended up disproving a 40-year-old conjecture.

A new proof illuminates the hidden patterns that emerge when addition becomes impossible.

The deceptively simple Kakeya conjecture has bedeviled mathematicians for 50 years. A new proof of the conjecture in three dimensions illuminates a whole crop of related problems.

In work that has been 30 years in the making, mathematicians have proved a major part of a profound mathematical vision called the Langlands program.

The proof creates stricter limits on potential exceptions to the famous Riemann hypothesis.

Mathematicians have struggled to prove Falconer’s Conjecture, a simple, but far-reaching, hypothesis about the distances between points. They’re finally getting close.

Mathematicians have illuminated what sets of points can look like if the distances between them are all whole numbers.

On its surface, the Kakeya conjecture is a simple statement about rotating needles. But it underlies a wealth of mathematics.