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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.

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

Michael Freedman’s momentous 1981 proof of the four-dimensional Poincaré conjecture was on the verge of being lost. The editors of a new book are trying to save it.

How readers used their geometry skills to survive a dangerous puzzle.

Using high school algebra and geometry, and knowing just one rational point on a circle or elliptic curve, we can locate infinitely many others.

Laurent Fargues and Peter Scholze have found a new, more powerful way of connecting number theory and geometry as part of the sweeping Langlands program.

A group of mathematicians has shown that at critical moments, a symmetry called rotational invariance is a universal property across many physical systems.

In this month’s puzzle, math is a question of life or death.

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