## Latest Articles

### To Pack Spheres Tightly, Mathematicians Throw Them at Random

Four mathematicians broke a 75-year-old record by finding a denser way to pack high-dimensional spheres.

### 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.

### Machine Learning Becomes a Mathematical Collaborator

Two recent collaborations between mathematicians and DeepMind demonstrate the potential of machine learning to help researchers generate new mathematical conjectures.

### Mathematicians Transcend Geometric Theory of Motion

More than 30 years ago, Andreas Floer changed geometry. Now, two mathematicians have finally figured out how to extend his revolutionary perspective.

### Mathematicians Find Long-Sought Building Blocks for Special Polynomials

Hilbert’s 12th problem asked for novel analogues of the roots of unity, the building blocks for certain number systems. Now, over 100 years later, two mathematicians have produced them.

### How Mathematicians Use Homology to Make Sense of Topology

Originally devised as a rigorous means of counting holes, homology provides a scaffolding for mathematical ideas, allowing for a new way to analyze the shapes within data.

### Mathematicians Settle Erdős Coloring Conjecture

Fifty years ago, Paul Erdős and two other mathematicians came up with a graph theory problem that they thought they might solve on the spot. A team of mathematicians has finally settled it.

### Mathematicians Set Numbers in Motion to Unlock Their Secrets

A new proof demonstrates the power of arithmetic dynamics, an emerging discipline that combines insights from number theory and dynamical systems.

### An Infinite Universe of Number Systems

The p-adics form an infinite collection of number systems based on prime numbers. They’re at the heart of modern number theory.