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Strangely Curved Shapes Break 50-Year-Old Geometry Conjecture

Mathematicians have disproved a major conjecture about the relationship between curvature and shape.

More than half a century later, Bruè, along with Aaron Naber of Northwestern University and Daniele Semola of the Swiss Federal Institute of Technology Zurich, would prove him wrong. If you impose some additional constraints — assuming, for instance, that you’re always working with a manifold that’s closed and bounded, like a sphere, or whose volume grows at a particular rate — Milnor’s conjecture holds in all dimensions. But it turns out that shapes with nonnegative Ricci curvature are more flexible and less well behaved than mathematicians had expected — complicating their understanding of the relationship between local geometric properties and global topological ones.

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