Mathematician unleashes 'a wave of new results' in geometric analysis

Mathematician unleashes 'a wave of new results' in geometric analysis: "If you drive into a parking garage and go up a level, that spiral ramp is part of a helicoid," he says. "And one of the things we were able to show was that every embedded minimal surface could be built out of these things. So the minimal surface either looks like a nice flat thing where the area is bounded, or it looks exactly like one of these double spiral staircases..."

They have been able to prove, he says, that of the infinite number of singularities that could possibly affect a surface through this curvature flow, only two types are stable enough to survive in reality. "If you were to wiggle your surface ever so slightly, in fact only two are stable of the infinitely many that are possible," Minicozzi says. "So if you are trying to understand [mean curvature flow] and you have to deal with all of these cases, it's much better to have to deal with two cases than an infinite number."

Shrinking Blob Computes Travelling Salesman Solutions

Shrinking Blob Computes Travelling Salesman Solutions:  In simplified terms, the blob clings to the dots as it shrinks, linking them with a minimal surface, rather like a soap bubble surface. “As the blob shrinks it morphologically adapts to the configuration of the cities,” they say...

The magic ingredient in all this is the special goo. It consists of many particles that each move according to a set of simple rules, like autonomous agents. These sit in a sea of “chemoattractant”, a virtual scent that the particles are attracted to.  At each stage in the calculation, each particle senses the chemoattractant around it and then moves towards the region of highest concentration. As it moves, it leaves behind its own trace of the chemoattractant for other particles to follow.

Simple mathematical pattern describes shape of neuron ‘jungle’ | KurzweilAI

Simple mathematical pattern describes shape of neuron ‘jungle’ | KurzweilAI: Cajal proposed that neurons spread out their branches so as to use as little wiring as possible to reach other cells in the network...

New work by UCL neuroscientists has revisited this century-old hypothesis using modern computational methods. They show that a simple computer program that connects points with as little wiring as possible can produce tree-like shapes that are indistinguishable from real neurons...

They also show that the shape of neurons follows a simple mathematical relationship called a power law*: dendrites grow to fill a target space in an optimal manner and, similar to a minimum spanning tree, use the least amount of wiring to reach all synaptic contacts.

Soap bubbles offer key to maximizing efficiency

Soap bubbles offer key to maximizing efficiency: "'It's more of a geometric issue because a sphere is the least-area way that you can enclose that amount of air,' said Morgan, who has published six books and more than 100 scientific articles. In 1999, he and his collaborators proved the 'Double Bubble Conjecture.'
The double bubble theorem effectively states that the double bubble 'provides the least-perimeter way to enclose and separate two prescribed volumes,' according to a paper Morgan co-authored that was published in 2004."