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# group theory

## Latest Articles

### Monumental Proof Settles Geometric Langlands Conjecture

In work that has been 30 years in the making, mathematicians have proved a major part of a profound mathematical vision called the Langlands program.

### Mathematicians Marvel at ‘Crazy’ Cuts Through Four Dimensions

Topologists prove two new results that bring some order to the confoundingly difficult study of four-dimensional shapes.

### She Finds the Poetry in Math and the Math in Poetry

The links between math, music and art have been explored for thousands of years. Sarah Hart is now turning a mathematical eye to literature.

### A Close-Up View Reveals the ‘Melting’ Point of an Infinite Graph

Just as ice melts to water, graphs undergo phase transitions. Two mathematicians showed that they can pinpoint such transitions by examining only local structure.

### Behold Modular Forms, the ‘Fifth Fundamental Operation’ of Math

Modular forms are one of the most beautiful and mysterious objects in mathematics. What are they?

### 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.