Harmonic analysis

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.

A new proof breaks a decades-long drought of progress on the problem of estimating the size of triangles created by cramming points into a square.

A new result shows how much of the plane can be colored by points that are never exactly one unit apart.

A new proof marks major progress toward solving the Kakeya conjecture, a deceptively simple question that underpins a tower of conjectures.

With her homeland mired in war, the sphere-packing number theorist Maryna Viazovska has become the second woman to win a Fields Medal in the award’s 86-year history.

Lillian Pierce wants to transform access to the world of mathematics, while making headway on problems that bridge the discrete and continuous.