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# Mathematics

## Latest Articles

### Surprising Limits Discovered in Quest for Optimal Solutions

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

### Where Transcendental Numbers Hide in Everyday Math

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

### How Tadayuki Watanabe Disproved a Major Conjecture About Spheres

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.

### How Wavelets Allow Researchers to Transform, and Understand, Data

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

### Mathematicians Prove Melting Ice Stays Smooth

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.

### In Topology, When Are Two Shapes the Same?

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.

### The Simple Math Behind the Mighty Roots of Unity

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.

### Mathematician Answers Chess Problem About Attacking Queens

The *n*-queens problem is about finding how many different ways queens can be placed on a chessboard so that none attack each other. A mathematician has now all but solved it.

### How Ancient War Trickery Is Alive in Math Today

Legend says the Chinese military once used a mathematical ruse to conceal its troop numbers. The technique relates to many deep areas of modern math research.