Category Archives: Math

Eigenvectors and Eigenvalues

Source: Quanta, Nov 2019

The physicists — Stephen Parke of Fermi National Accelerator Laboratory, Xining Zhang of the University of Chicago and Peter Denton of Brookhaven National Laboratory — had arrived at the mathematical identity about two months earlier while grappling with the strange behavior of particles called neutrinos.

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.

Eigenvectors and eigenvalues are independent, and normally they must be calculated separately starting from the rows and columns of the matrix itself. College students learn how to do this for simple matrices. But the new formula differs from existing methods. “What is remarkable about this identity is that at no point do you ever actually need to know any of the entries of the matrix to work out anything,” said Tao.

The formula makes sense in hindsight, Tao said, because the eigenvalues of the minor matrix encode hidden information. But “it certainly was not something that I, for instance, would have thought about.”

It’s unusual in mathematics for a tool to appear that’s not already associated with a problem, he said. But he thinks the relationship between eigenvectors and eigenvalues is bound to matter. “It’s so pretty that I’m sure it will have some use in the near future,” he said. “Right now, we just have one application.”

Experts say that more applications might arise, since so many problems involve calculating eigenvectors and eigenvalues. “This is of really broad applicability,” said John Beacom, a particle physicist at Ohio State University. “Who knows what doors it will open.”

Pure mathematicians feel similarly. “This is certainly both surprising and interesting,” said Van Vu, a mathematician at Yale University. “I did not suspect that one can compute eigenvectors using only information about eigenvalues.”

it’s not surprising that a new insight into centuries-old mathematical objects came from physicists. Nature has inspired mathematical thinking ever since humans started counting on 10 fingers. “For math to thrive, it has to connect to nature,” Vu said. “There is no other way.”

The expressions for the eigenvalues are simpler than those of eigenvectors, so Parke, Zhang and Denton started there. Previously, they had developed a new method to closely approximate the eigenvalues. With these in hand, they noticed that the long eigenvector expressions seen in previous works were equal to combinations of those eigenvalues. Putting the two together, “you can calculate neutrino oscillations in matter fast and simply,” Zhang said.

As for how they spotted the pattern that suggested the formula, the physicists aren’t sure. Parke said they simply noticed instances of the pattern and generalized. He admits to being good at solving puzzles. In fact, he is credited with co-discovering another important pattern in 1986 that has streamlined particle physics calculations and inspired discoveries ever since.

Still, the fact that the strange behavior of neutrinos could lead to new insights about matrices came as a shock. “People have been solving problems with linear algebra for a very, very long time,” Parke said. “I’m expecting someday to get an email from somebody that says, ‘If you look at this obscure paper by [the 19th-century mathematician] Cauchy, in the third appendix in a footnote, it’s there.’”

Related Resource: Terry Tao blog, Nov 2019

Thanks for this link! Yes, this does seem to be the same identity (and the author here specifically states that it has been discovered a couple times in the past but never successfully publicised to a wider audience). As far as I can tell this particular preprint was never published either – it seems for some reason that this particular identity just has a tough time trying to become known in the literature.

(It’s at times like this that I wish we had a good semantic search engine for mathematics – I am not sure how I would have discovered this preprint other than by publishing our own preprint and waiting for someone to notice the connection.)

Terry Tao Interview

Source: Princeton, Nov 2019

erence Tao *96’s book, Solving Mathematical Problems: A Personal Perspective, is an engagingly slender volume, full of insights on how to approach problems in number theory, algebra, Euclidean geometry, and analytic geometry.

Tao began by setting out some sensible strategies for problem-solving, including these: Understand the problem, understand the data, understand the objective, select good notation, and write down everything you know. He also hoped for something less rote. “A solution,” Tao proposed, “should be relatively short, understandable, and hopefully have a touch of elegance. It should be fun to discover.”

Tao wrote Solving Mathematical Problems in 1990, when he was 15 years old.

The process of problem-solving, he emphasizes, is “non-linear.” In the end, though, is mathematics — is the universe — orderly or random? Tao warms to the question.

“It depends on where you look,” he says. “At the extremely microscopic level, the laws of nature are ordered. Particles and quantum waves obey very rigid waves of mechanics. But as you go to more complicated objects, molecules and living creatures, then it becomes more chaotic and unpredictable.

“There’s this weird mathematical phenomenon called universality. You get very complicated systems, of atoms or people, but if you look at it at a large-enough scale, order starts emerging. Einstein once said that the most incomprehensible thing about the universe is that it is comprehensible. It is very complicated, but at certain levels, patterns appear again.

“So there is order — sometimes — but there is also chaos.” 

Cutting Shapes to Make Equations

Source: Quanta, Oct 2019

If you have two flat paper shapes and a pair of scissors, can you cut up one shape and rearrange it as the other? If you can, the shapes are “scissors congruent.”

But, mathematicians wonder, can you tell if two shapes share this relationship even without using scissors? In other words, are there characteristics of each shape you could measure ahead of time to determine whether they’re scissors congruent?

For two-dimensional shapes, the answer is easy: Just determine their areas. If they’re the same, the shapes are scissors congruent.

But for higher-dimensional shapes — like a three-dimensional ball, or an 11-dimensional doughnut that’s impossible to picture — the question of cutting one up and reassembling it as another is a lot harder. Despite centuries of effort, mathematicians have been unable to identify characteristics that dictate scissors congruence for most higher-dimensional shapes.

Ramsey Theory: Order from Disorder

Source: Quanta, May 2016

A Great Mathematical Proof

Source: RJ Lipton blog, Oct 2019

Proofs in mathematics and not just formal arguments that show that a theorem is correct. They are much more. They must show why and how something is true.

They must explain and extend our understanding of why something is true. They must do more than just demonstrate that something is correct.

They must also make it clear what they claim to prove.

A difficulty we felt, then, was that care must be given to what one is claiming to prove. In mathematics often what is being proved is simple to state. In practice that is less clear. A long complex statement may not correctly capture what one is trying to prove.

a great proof is a proof that helps create new proofs of something else.

What mean is a great proof is one that enables new insights, that enables further progress, that advances the field. Not just a result that “checks” for correctness.

Equivalence Shakes Mathematics

Source: Quanta, Oct 2019

there is a growing community of mathematicians who regard the equal sign as math’s original error. They see it as a veneer that hides important complexities in the way quantities are related — complexities that could unlock solutions to an enormous number of problems. They want to reformulate mathematics in the looser language of equivalence.

“We came up with this notion of equality,” said Jonathan Campbell of Duke University. “It should have been equivalence all along.”

the growing pains that a venerable field like mathematics undergoes whenever it tries to absorb a big new idea, especially an idea that challenges the meaning of its most important concept. “There’s an appropriate level of conservativity in the mathematics community,” said Clark Barwick of the University of Edinburgh. “I just don’t think you can expect any population of mathematicians to accept any tool from anywhere very quickly without giving them convincing reasons to think about it.”

While equality is a strict relationship — either two things are equal or they’re not — equivalence comes in different forms.

When you can exactly match each element of one set with an element in the other, that’s a strong form of equivalence. But in an area of mathematics called homotopy theory, for example, two shapes (or geometric spaces) are equivalent if you can stretch or compress one into the other without cutting or tearing it.

From the perspective of homotopy theory, a flat disk and a single point in space are equivalent — you can compress the disk down to the point. Yet it’s impossible to pair points in the disk with points in the point. After all, there’s an infinite number of points in the disk, while the point is just one point.

A category is a set with extra metadata: a description of all the ways that two objects are related to one another, which includes a description of all the ways two objects are equivalent.

In the perspective of category theory, you forget about the explicit way in which any one object is described and focus instead on how an object is situated among all other objects of its type.

In these subtler notions of equivalence, the amount of information about how two objects are related increases dramatically.

Two triangles are homotopy equivalent if you can stretch or otherwise deform one into the other. Two points on the surface are homotopy equivalent if there’s a path linking one with the other. By studying homotopy paths between points on the surface, you’re really studying different ways in which the triangles represented by those points are related.

But it’s not enough to say that two points are linked by many equal paths. You need to think about equivalences between all those paths, too. So in addition to asking whether two points are equivalent, you’re now asking whether two paths that start and end at the same pair of points are equivalent — whether there’s a path between those paths. This path between paths takes the shape of a disk whose boundary is the two paths.

Lurie’s work represented a big challenge. At heart it was a provocation: Here is a better way to do math. The message was especially pointed for mathematicians who’d spent their careers developing methods that Lurie’s work transcended.

Lurie’s work was hard to swallow in other ways. The volume of material meant that mathematicians would need to invest years reading his books. That’s an almost impossible requirement for busy mathematicians in midcareer, and it’s a highly risky one for graduate students who have only a few years to produce results that will get them a job.

Lurie’s work was also highly abstract, even in comparison with the highly abstract nature of everything else in advanced mathematics. As a matter of taste, it just wasn’t for everyone. “Many people did view Lurie’s work as abstract nonsense, and many people absolutely loved it and took to it,” Campbell said. “Then there were responses in between, including just full-on not understanding it at all.”

The inaccessibility of Lurie’s books has led to an imprecision in some of the subsequent research based on them. Lurie’s books are hard to read, they’re hard to cite, and they’re hard to use to check other people’s work.

Riehl and Verity hope to move infinity category theory forward in another way as well. They’re specifying aspects of infinity category theory that work regardless of the model you’re in. This “model-independent” presentation has a plug-and-play quality that they hope will invite mathematicians into the field who might have been staying away while Higher Topos Theory was the only way in.

Translating Equations into Geometry

Source: Quanta, Sep 2019