Unexpectedly Intriguing!
24 December 2019

If 2018 was the year of amateurs advancing maths via social media, 2019 was the year an unexpected connection via social media led to a discovery of a basic math identity.

The criteria we use in selecting the stories that make our year-end list is that the maths they involve must either resolve long-standing questions or have real practical applications. Not all of it comes from well trodden corners of mathematics, which is why we try to pair the stories with links to additional background material or videos to help explain the math that's behind the story.

That's enough of a set up for this year. Let's get straight to it, shall we? It all begins below....

Suppose that you need to multiply two really, really big numbers together using the long multiplication method you learned back in grade school. The one where you systematically multiply each individual digit in one number by every digit in another, keeping a rolling tally where you carry numbers to be added to the next digit as you work your way from right to left in performing, pardon the pun, a multitude of multiplications. Wouldn't it be nice if you could get to the product with less work?

David Harvey and Joris van der Hoeven have developed a method using multidimensional Fast Fourier Transforms, which have practical application in the digital signal processing used to support modern telecommunications, including the compression methods used for digital images, audio, and video files that make it possible for millions to enjoy the millions of cat videos on the Internet.

What Harvey and van der Hoeven have wrought is a method that makes it possible to greatly reduce the number of individual operations that would otherwise have to take place when two large numbers are multiplied together. Traditionally, multiplying two 10,000-digit long numbers would require 100,000,000 individual operations to complete. In their new method, you would require only 92,104 operations to perform the multiplication, a 99.9% reduction.

That kind of reduction offers immense practical value in improving the efficiency and reducing the time needed to perform similar operations using today's established calculation methods.

Although the name of our blog is Political Calculations, we strive to be generally apolitical, where we much prefer to focus on how math and politics intersect.

2019 was a tailor-made year for us in that the very definition of what constitutes a political majority became a major issue in the South American nation of Guyana, which ultimately required a court of law to uphold the mathematical order of operations to resolve.

In doing so, a no-confidence motion was upheld in the Guyanese legislature, which led to the fall of that nation's parliamentary government. The political side of that story is still playing out, with the nation set to hold elections to seat a new government on 2 March 2020, as demanded by better algebra.

Still, though it's not yet done, that counts as a resolution. The other notable story involving the mathematical order of operations in 2019 had to do with a controversial Twitter post featuring the mathematical phrasing equivalent of the sentence "she fed her cat food", with more than one way of parsing the meaning of the ambiguous phrasing. We're not going back down that rabbit hole, because it's too political!

The biggest math story of 2017 involved the discovery of what may be the limits of the Navier-Stokes equations that describe the flow of fluids, which if and when proven, will almost certainly claim one of the million-dollar millennium prizes offered by the Clay Mathematics Institute. 2019 has a similar, but much smaller scale and prize-free version of that story, involving when and how some of the pioneering fluid flow equations developed centuries ago by Leonhard Euler might blow up.

And blow up they have, where a new proof by Tarek Elgindi that was followed up by Elgindi with Tej-eddine Ghoul and Nader Masmoudi. Quanta Magazine's Kevin Hartnett explains:

Elgindi’s work is not a death knell for the Euler equations. Rather, he proves that under a very particular set of circumstances, the equations overheat, as it were, and start to output nonsense. Under more realistic conditions, the equations are still, for now, invulnerable.

But the exception Elgindi found is startling to mathematicians, because it occurred under conditions where they previously thought the equations always functioned.

In much the same way we still use the equations developed by Isaac Newton, even though we have Einstein's math to cope with more extreme conditions, we'll keep using Euler's fluid equations because they reasonably approximate a good portion of what we observe in the real world. What's changed is that we now know under what conditions those equations stop reasonably approximating reality, which tells us when and where we will need to use other mathematical tools.

That's useful knowledge, but not quite the biggest math story of the year.

Perhaps the most extraordinary math story of the year came when a team of particle physicists, Stephen Parke, Xining Zhang, and Peter Denton, had a question about linear algebra they asked on Reddit roughly five months ago, after they found a simple relationship between eigenvalues and eigenvectors that seemed to allow them to use relatively easy-to-work-with eigenvalues in place of eigenvectors in their analysis of neutrino behavior, which they were looking to validate.

Quanta's Natalie Wolchover describes their discovery:

They’d noticed that hard-to-compute terms called “eigenvectors,” describing, in this case, the ways that neutrinos propagate through matter, were equal to combinations of terms called “eigenvalues,” which are far easier to compute. Moreover, they realized that the relationship between eigenvectors and eigenvalues — ubiquitous objects in math, physics and engineering that have been studied since the 18th century — seemed to hold more generally.

Although the physicists could hardly believe they’d discovered a new fact about such bedrock math, they couldn’t find the relationship in any books or papers. So they took a chance and contacted Tao, despite a note on his website warning against such entreaties.

The ensuing discussion pointed them toward previous work by Terrence Tao, who confirmed the result and generated no fewer than three separate proofs confirming the relationship. Digging deeper into the literature, the physicists and Tao found that parts of the relationship they had identified had previously been developed as long ago as 1966, but no one had quite put all the pieces together in the way they had.

But now they have, with the result that they've created an enduring basic math identity that already is being put to practical use.

That's why the tale of how research into the behavior of neutrinos led to the confirmation of a basic math identity in linear algebra is the biggest math story of the year.

### Previously on Political Calculations

The Biggest Math Story of the Year is how we've traditionally marked the end of our posting year since 2014. Here are links to our previous editions, along with our coverage of other math stories during 2019:

It's not your imagination. We covered more stories about math in 2019 than in any previous year!

This is Political Calculations final post for 2019. Thank you for passing time with us this year, have a Merry Christmas and a wonderful holiday season, and we'll see you again in the New Year, where there are rumblings that the Inventions In Everything series will return!

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