Category Archives: Math

Quanta Highlights Math Achievements

Sources: Quanta, 2019

https://www.quantamagazine.org/with-category-theory-mathematics-escapes-from-equality-20191010/

Lurie’s books are the single, authoritative text on infinity categories. They are completely rigorous, but hard to completely grasp. They’re especially poorly suited to serving as reference manuals — it’s difficult to look up specific theorems, or to check that a specific application of infinity categories that one might encounter in someone else’s paper really works out.

https://www.quantamagazine.org/foundations-built-for-a-general-theory-of-neural-networks-20190131/

Neural networks aim to mimic the human brain — and one way to think about the brain is that it works by accreting smaller abstractions into larger ones. Complexity of thought, in this view, is then measured by the range of smaller abstractions you can draw on, and the number of times you can combine lower-level abstractions into higher-level abstractions — like the way we learn to distinguish dogs from birds.

power in taking small pieces and combining them at greater levels of abstraction instead of attempting to capture all levels of abstraction at once.

https://www.quantamagazine.org/random-surfaces-hide-an-intricate-order-20190702/

Mathematicians have proved that a random process applied to a random surface will yield consistent patterns.

https://www.quantamagazine.org/how-artificial-intelligence-is-changing-science-20190311/

generative modeling asks how likely it is, given condition X, that you’ll observe outcome Y.

Cognitive Technologies … Sentences –> Symbols

Source: Dominic Cummings blog, Jun 2019

Language and writing were cognitive technologies created thousands of years ago which enabled us to think previously unthinkable thoughts. Mathematical notation did the same over the past 1,000 years. For example, take a mathematics problem described by the 9th Century mathematician al-Khwarizmi (who gave us the word algorithm):

Once modern notation was invented, this could be written instead as:

x2 + 10x = 39

Michael Nielsen uses a similar analogy. Descartes and Fermat demonstrated that equations can be represented on a diagram and a diagram can be represented as an equation. This was a new cognitive technology, a new way of seeing and thinking: algebraic geometry. Changes to the ‘user interface’ of mathematics were critical to its evolution and allowed us to think unthinkable thoughts (Using Artificial Intelligence to Augment Human Intelligence, see below).

Similarly in the 18th Century, there was the creation of data graphics to demonstrate trade figures. Before this, people could only read huge tables. This is the first data graphic:

screenshot 2019-01-29 00.28.21

The Jedi of data visualisation, Edward Tufte, describes this extraordinary graphic of Napoleon’s invasion of Russia as ‘probably the best statistical graphic ever drawn’. It shows the losses of Napoleon’s army: from the Polish-Russian border, the thick band shows the size of the army at each position, the path of Napoleon’s winter retreat from Moscow is shown by the dark lower band, which is tied to temperature and time scales (you can see some of the disastrous icy river crossings famously described by Tolstoy). NB. The Cabinet makes life-and-death decisions now with far inferior technology to this from the 19th Century (see below).

screenshot 2019-01-29 10.37.05

If we look at contemporary scientific papers they represent extremely compressed information conveyed through a very old fashioned medium, the scientific journal. Printed journals are centuries old but the ‘modern’ internet versions are usually similarly static. They do not show the behaviour of systems in a visual interactive way so we can see the connections between changing values in the models and changes in behaviour of the system. There is no immediate connection. Everything is pretty much the same as a paper and pencil version of a paper. In Media for Thinking the Unthinkable, Victor shows how dynamic tools can transform normal static representations so systems can be explored with immediate feedback. This dramatically shows how much more richly and deeply ideas can be explored. With Victor’s tools we can interact with the systems described and immediately grasp important ideas that are hidden in normal media.

Circles Disturbed

Source: Princeton University Press, 2012

Circles Disturbed delves into topics such as the way in which historical and biographical narratives shape our understanding of mathematics and mathematicians, the development of “myths of origins” in mathematics, the structure and importance of mathematical dreams, the role of storytelling in the formation of mathematical intuitions, the ways mathematics helps us organize the way we think about narrative structure, and much more.

Related Resource: JStor, date indeterminate

CHAPTER 1 From Voyagers to Martyrs: Toward a Storied History of Mathematics

 (pp. 1-51)

AMIR ALEXANDER
CHAPTER 6 Visions, Dreams, and Mathematics

(pp. 183-210)

BARRY MAZUR

Mathematicians can hardly avoid making use of stories of various kinds, to say nothing of images, sketches, and diagrams, to help convey the meaning of their accomplishments and their aims. As Peter Galison points out in chapter 2, we mathematicians often are nevertheless silent—or perhaps even uneasy—about the role that stories and images play in our work.

If someone asks usWhat is X?, whereXis some mathematical concept, we boldly answer, for we have been well trained in the art of definition. All the fine articulations of logical structure are at our fingertips. If, however, someone…

CHAPTER 8 Mathematics and Narrative: Why Are Stories and Proofs Interesting?

(pp. 232-243)

BERNARD TEISSIER

There are many types of narrations, from origin myths to the ship logs of maritime explorers, from children’s bedtime stories to works of literature—including poetry—and theater. We might also recall here Kipling’s joke in one of his letters from Japan about the person who, having borrowed a dictionary, gives it back with the comment that the stories are generally interesting, but too diverse.

The concept of narration varies with location and is not easy to define. Is a haiku a narration? Is Heraclitus’spanta rhei—“all things are flowing”—the concise narration of a part of his experience…

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