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# group theory

## Latest Articles

### 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.