Mathematics

In nonreciprocal systems, where Newton’s third law falls apart, “exceptional points” are helping researchers understand phase transitions and possibly other phenomena.

New work establishes a tighter connection between the rank of a polynomial and the extent to which it favors particular outputs.

Algorithms that zero in on solutions to optimization problems are the beating heart of machine reasoning. New results reveal surprising limits.

The transcendental number π is as familiar as it is ubiquitous, but how does Euler’s number e transcend the ordinary?

Watanabe invented a new way of distinguishing shapes on his way to solving the last open case of the Smale conjecture, a central question in topology about symmetries of the sphere.

Built upon the ubiquitous Fourier transform, the mathematical tools known as wavelets allow unprecedented analysis and understanding of continuous signals.

After decades of effort, mathematicians now have a complete understanding of the complicated equations that model the motion of free boundaries, like the one between ice and water.

As topologists seek to classify shapes, the effort hinges on how to define a manifold and what it means for two of them to be equivalent.

Solutions to the simplest polynomial equations — called “roots of unity” — have an elegant structure that mathematicians still use to study some of math’s greatest open questions.