Fractals

In the late 19th century, Karl Weierstrass invented a fractal-like function that was decried as nothing less than a “deplorable evil.” In time, it would transform the foundations of mathematics.

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

Simple rules in simple settings continue to puzzle mathematicians, even as they devise intricate tools to analyze them.

For decades, a small group of mathematicians has patiently unraveled the mystery of what was once math’s most popular picture. Their story shows how technology transforms even the most abstract mathematical landscapes.

A new paper establishes a long-conjectured bound about the size of the overlap between sets of lines and points.

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

Mark Braverman has spent his career translating thorny problems into the language of information complexity.

The concept of dimension seems simple enough, but mathematicians struggled for centuries to precisely define and understand it.

A new proof demonstrates the power of arithmetic dynamics, an emerging discipline that combines insights from number theory and dynamical systems.