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A pair of researchers has shown that trying to classify groups of numbers called “torsion-free abelian groups” is as hard as it can possibly be.
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.
For 50 years, mathematicians have believed that the total number of real numbers is unknowable. A new proof suggests otherwise.
A group of mathematicians has shown that at critical moments, a symmetry called rotational invariance is a universal property across many physical systems.
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