Welcome to the Quanta Audio Edition.

In each of these bi-weekly episodes, we bring you a story direct from the Quanta website

about developments in basic science and mathematics.

I'm Susan Vallett.

Researchers use a powerful mathematical technique to model melting ice and other phenomena,

but it's long been plagued by certain nightmare scenarios that cause these mathematical descriptions to break down.

Now, a new proof has removed that obstacle. That's next.

Check out this feed every Tuesday for the Quanta podcast.

That's where editor-in-chief Samir Patel talks to our writers and editors about more of Quanta's most popular, interesting, and thought-provoking stories. story. Imagine an ice cube floating in a glass of water.

Eventually it will melt down to a tiny frozen speck before disappearing.

As it shrinks, its surface gets smoother and any irregularities or sharp edges gradually vanish.

Mathematicians want to understand this process in greater detail to be able to say exactly how the

surface of the ice, or, say, the shape of a gradually eroding sandcastle changes over time.

To analyze this phenomenon, they study how more abstract mathematical surfaces and shapes evolve

according to a particular set of rules.

This set of rules defines a process called mean curvature flow,

which simultaneously smooths out a surface, even a highly irregular

one, and shrinks it. But as the surface evolves, singularities can form, points where our

mathematical descriptions break down. The surface might jut out sharply, or it might thin to a point where the curvature blows up to infinity.

Many common kinds of surfaces, such as those that are closed up, like a sphere, are guaranteed to exhibit singularities during mean curvature flow.

If these singularities are too complicated, it becomes impossible for the flow to continue.

Mathematicians want to ensure that even after a singularity forms, they can still analyze how

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