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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.

So-called topological quantum computing would avoid many of the problems that stand in the way of full-scale quantum computers. But high-profile missteps have led some experts to question whether the field is fooling itself.

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.

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.

Researchers are turning to the mathematics of higher-order interactions to better model the complex connections within their data.

Mathematicians using the computer program Lean have verified the accuracy of a difficult theorem at the cutting edge of research mathematics.

Even in an incomplete state, quantum field theory is the most successful physical theory ever discovered. Nathan Seiberg, one of its leading architects, talks about the gaps in QFT and how mathematicians could fill them.

In three towering papers, a team of mathematicians has worked out the details of Liouville quantum field theory, a two-dimensional model of quantum gravity.

The accelerating effort to understand the mathematics of quantum field theory will have profound consequences for both math and physics.

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