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# Partial differential equations

## Latest Articles

### A Tower of Conjectures That Rests Upon a Needle

On its surface, the Kakeya conjecture is a simple statement about rotating needles. But it underlies a wealth of mathematics.

### New Proof Threads the Needle on a Sticky Geometry Problem

A new proof marks major progress toward solving the Kakeya conjecture, a deceptively simple question that underpins a tower of conjectures.

### Computer Proof ‘Blows Up’ Centuries-Old Fluid Equations

For more than 250 years, mathematicians have wondered if the Euler equations might sometimes fail to describe a fluid’s flow. A new computer-assisted proof marks a major breakthrough in that quest.

### Mathematicians Prove Melting Ice Stays Smooth

After decades of effort, mathematicians now have a complete understanding of the complicated equations that model the motion of free boundaries, like the one between ice and water.

### Mathematicians Identify Threshold at Which Shapes Give Way

A new proof establishes the boundary at which a shape becomes so corrugated, it can be crushed.

### Latest Neural Nets Solve World’s Hardest Equations Faster Than Ever Before

Two new approaches allow deep neural networks to solve entire families of partial differential equations, making it easier to model complicated systems and to do so orders of magnitude faster.

### In Violation of Einstein, Black Holes Might Have ‘Hair’

A new study shows that extreme black holes could break the famous “no-hair” theorem, and in a way that we could detect.

### Symbolic Mathematics Finally Yields to Neural Networks

After translating some of math’s complicated equations, researchers have created an AI system that they hope will answer even bigger questions.

### New Math Proves That a Special Kind of Space-Time Is Unstable

Einstein’s equations describe three canonical configurations of space-time. Now one of these three — important in the study of quantum gravity — has been shown to be inherently unstable.