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When two mathematicians raised pointed questions about a classic proof that no one really understood, they ignited a years-long debate about how much could be trusted in a new kind of geometry.

Three physicists have been awarded the Nobel Prize in Physics for rewriting our understanding of exotic quantum states on the surfaces of materials.

Mathematicians have had a hard time finding commonalities in large groups of random shapes — until recently.

Researchers have uncovered deep connections among different types of random objects, illuminating hidden geometric structures.

At 28, Peter Scholze is uncovering deep connections between number theory and geometry.

Coils and twirls in DNA’s double-helix change how the molecule behaves, opening a new role for topology in the study of life.

The Fields medalist Vladimir Voevodsky has died at 51. This 2015 article describes his computer-aided quest to eliminate human error and rewrite the century-old rules underlying all of mathematics.

Mathematicians have solved the century-old triangulation conjecture, a major problem in topology that asks whether all spaces can be subdivided into smaller units.

Researchers have found a set of rules for imbuing flat surfaces with curvature, enabling them to form a virtually unlimited range of three-dimensional structures.