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Two monumental works have led many mathematicians to avoid the equal sign. The process has not always gone smoothly.

A new proof shows why an uncountably infinite number of Möbius strips will never fit into a three-dimensional space.

As chemists tie the most complicated molecular knot yet, biophysicists create a “periodic table” that describes what kinds of knots are possible.

The 30-year-old math sensation Peter Scholze is now one of the youngest Fields medalists for “the revolution that he launched in arithmetic geometry.”

A complete classification could lead to a wealth of new materials and technologies. But some exotic phases continue to resist understanding.

By reimagining the kinks and folds of origami as atoms in a lattice, researchers are uncovering strange behavior hiding in simple structures.

Voevodsky’s friends remember him as constitutionally unable to compromise on the truth — a quality that led him to produce some of the most important mathematics of the 20th century.

It’s “a definitive study for all time, like writing the final book,” says one researcher who’s mapping out new classes of geometric structures.

Two “rare jewels” have illuminated a mysterious multidimensional object that connects a huge variety of mathematical work.

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