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Math Professor and Students Solve One of the World's Long-Standing Math Problems

The Caltech researchers took a "random walk" together.

Math Professor and Students Solve One of the World's Long-Standing Math Problems
The Caltech researchersCalifornia Institute of Technology

You may take an ambling walk in the countryside, or even around your neighborhood and call it random, however, in the world of math it is even more random than that. The long-standing math problem called "random walk" would mean that for each step that you take on your walk you would have to flip a coin to decide where to go. 

Mathematicians from the California Institute of Technology (Caltech) have finally solved this old math problem.

Random walk problem

The Caltech team consisted of Omer Tamuz, professor of mathematics and economics, as well as two of his students, Joshua Fritz and Pooya Vahidi Ferdowsi, and their colleague Yair Hartman from Ben-Gurion University in Israel. 

Tamuz explained "I remember talking to the students about a realization we had regarding this problem, and then the next morning I found out they had stayed up late into the night and figured it out."

Not an easy feat given that around 90% of what mathematicians work on doesn't get solved, according to Frischer. "Something like 90 percent of the projects you work on, you are not going to be able to solve. With about 10 percent, you start making progress and work much harder," said Frischer. 

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The term "random walk" is a colloquialism for what's known in math as a way to create a path based on random choices at different junctions. Random walks are not only used in math, however. Biologists use the theory to understand how animals move and behave, physicists use it to understand and describe how particles move, and computer scientists use it to create video games. 

Tamuz pointed out the difference in random walks: "Say you have two societies, and one of them makes some technological advancement while the other suffers a natural disaster. Are these differences going to persist forever, or will they eventually disappear and we'll forget that once there was an advantage?"

He continued "In random walks, it has been long known that there are groups that have these memories while in other groups the memories are erased. But it was not really clear which groups have this property and which don't—that is, what makes a group have memory? This is what we figured out."

What the team did in order to solve the problem was to combine algebra ideas with geometry ones when describing random walks. It's by using this connection that they were able to come to a conclusion. 

The Caltech researchers found that random walks that meet a certain standard based on vector geometry are the walks that converge with everything else. 

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