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New Proof Distinguishes Mysterious and Powerful ‘Modular Forms’
Using “refreshingly old” tools, mathematicians resolved a 50-year-old conjecture about how to categorize important functions called modular forms, with consequences for number theory and theoretical physics.
Probability and Number Theory Collide — in a Moment
Mathematicians are taking ideas developed to study random numbers and applying them to a broad range of categories.
Mathematicians Crack a Simple but Stubborn Class of Equations
Ever since Archimedes, mathematicians have been fascinated by equations that involve a difference between squares. Now two mathematicians have proven how often these equations have solutions, concluding a decades-old quest.
The Simple Math Behind the Mighty Roots of Unity
Solutions to the simplest polynomial equations — called “roots of unity” — have an elegant structure that mathematicians still use to study some of math’s greatest open questions.
Galois Groups and the Symmetries of Polynomials
By focusing on relationships between solutions to polynomial equations, rather than the exact solutions themselves, Évariste Galois changed the course of modern mathematics.
New Shape Opens ‘Wormhole’ Between Numbers and Geometry
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.
Mathematician Disproves 80-Year-Old Algebra Conjecture
Inside the symmetries of a crystal shape, a postdoctoral researcher has unearthed a counterexample to a basic conjecture about multiplicative inverses.
The ‘Useless’ Perspective That Transformed Mathematics
Representation theory was initially dismissed. Today, it’s central to much of mathematics.
The Map of Mathematics
Explore our surprisingly simple, absurdly ambitious and necessarily incomplete guide to the boundless mathematical universe.