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Mathematical Patterns in/of the Universe

Source: Quanta magazine, Feb 2013

Subatomic particles have little to do with decentralized bus systems. But in the years since the odd coupling was discovered, the same pattern has turned up in other unrelated settings. Scientists now believe the widespread phenomenon, known as “universality,” stems from an underlying connection to mathematics, and it is helping them to model complex systems from the Internet to Earth’s climate.

The red pattern exhibits a precise balance of randomness and regularity known as “universality,” which has been observed in the spectra of many complex, correlated systems. In this spectrum, a mathematical formula called the “correlation function” gives the exact probability of finding two lines spaced a given distance apart.

Universality is thought to arise when a system is very complex, consisting of many parts that strongly interact with each other to generate a spectrum. The pattern emerges in the spectrum of a random matrix, for example, because the matrix elements all enter into the calculation of that spectrum. But random matrices are merely “toy systems” that are of interest because they can be rigorously studied, while also being rich enough to model real-world systems, Vu said. Universality is much more widespread. Wigner’s hypothesis (named after Eugene Wigner, the physicist who discovered universality in atomic spectra) asserts that all complex, correlated systems exhibit universality, from a crystal lattice to the Internet.

The more complex a system is, the more robust its universality should be, said László Erdös of the University of Munich, one of Yau’s collaborators. “This is because we believe that universality is the typical behavior.”

“It may happen that it is not a matrix that lies at the core of both Wigner’s universality and the zeta function, but some other, yet undiscovered, mathematical structure,” Erdös said. “Wigner matrices and zeta functions may then just be different representations of this structure.”

Many mathematicians are searching for the answer, with no guarantee that there is one. “Nobody imagined that the buses in Cuernavaca would turn out to be an example of this. Nobody imagined that the zeroes of the zeta function would be another example,” Dyson said. “The beauty of science is it’s completely unpredictable, and so everything useful comes out of surprises.”

Peter Scholze: deep connections between number theory and geometry.

Source: Quanta, Jun 2016

The 22-year-old student, Peter Scholze, had found a way to sidestep one of the most complicated parts of the proof, which deals with a sweeping connection between number theory and geometry.

his unnerving ability to see deep into the nature of mathematical phenomena. Unlike many mathematicians, he often starts not with a particular problem he wants to solve, but with some elusive concept that he wants to understand for its own sake.

As Scholze burrowed into the proof, he became captivated by the mathematical objects involved — structures called modular forms and elliptic curves that mysteriously unify disparate areas of number theory, algebra, geometry and analysis. Reading about the kinds of objects involved was perhaps even more fascinating than the problem itself, he said.

Scholze’s mathematical tastes were taking shape. Today, he still gravitates toward problems that have their roots in basic equations about whole numbers. Those very tangible roots make even esoteric mathematical structures feel concrete to him. “I’m interested in arithmetic, in the end,” he said. He’s happiest, he said, when his abstract constructions lead him back around to small discoveries about ordinary whole numbers.

Scholze avoids getting tangled in the jungle vines by forcing himself to fly above them: As when he was in college, he prefers to work without writing anything down. That means that he must formulate his ideas in the cleanest way possible, he said. “You have only some kind of limited capacity in your head, so you can’t do too complicated things.”

In the middle of the 20th century, mathematicians discovered an astonishing link between reciprocity laws and what seemed like an entirely different subject: the “hyperbolic” geometry of patterns such as M.C. Escher’s famousangel-devil tilings of a disk. This link is a core part of the “Langlands program,” a collection of interconnected conjectures and theorems about the relationship between number theory, geometry and analysis. When these conjectures can be proved, they are often enormously powerful: For instance, the proof of Fermat’s Last Theorem boiled down to solving one small (but highly nontrivial) section of the Langlands program.

Mathematicians have gradually become aware that the Langlands program extends far beyond the hyperbolic disk; it can also be studied in higher-dimensional hyperbolic spaces and a variety of other contexts. Now, Scholze has shown how to extend the Langlands program to a wide range of structures in “hyperbolic three-space” — a three-dimensional analogue of the hyperbolic disk — and beyond. By constructing a perfectoid version of hyperbolic three-space, Scholze has discovered an entirely new suite of reciprocity laws.

Soap Foam as a Rose – Kanebo Evita

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More information available: http://www.kanebo-cosmetics.jp/evita/

 

Ramanujan’s Puzzles

Source: Quanta Magazine, Jul 2016

The work that Ramanujan did in his brief professional life a century ago has spawned whole new areas of mathematical investigation, kept top mathematicians busy for their whole professional lives, and is finding applications in computer science, string theory, and the mathematical basis of black hole physics.

The mathematician Mark Kac divided all geniuses into two types: “ordinary” geniuses, who make you feel that you could have done what they did if you were say, a hundred times smarter, and “magical geniuses,” the working of whose minds is, for all intents and purposes, incomprehensible. There is no doubt that Srinivas Ramanujan was a magical genius, one of the greatest of all time. Just looking at any of his almost 4,000 original results can inspire a feeling of bewilderment and awe even in professional mathematicians: What kind of mind can dream up exotic gems like these?

Ramanujan indeed had preternatural insights into infinity: he was a consummate bridge builder between the finite and the infinite, finding ways to represent numbers in the form of infinite series, infinite sums and products, infinite integrals, and infinite continued fractions, an area in which, in the words of Hardy, his mastery was “beyond that of any mathematician in the world.”

What if Ramanujan had modern calculating tools?

Ramanujan, like many other great mathematical geniuses such as Gauss, loved to play with specific cases, which he then built into general results. These geniuses did prodigious calculations by hand — Ramanujan used chalk on slate in his early days, erasing intermediate results with his elbow.

Steven Wolfram has conjectured that if Ramanujan had modern calculating tools like Mathematica, “he would have been quite an adventurergoing out into the mathematical universe and finding all sorts of strange and wonderful things, then using his intuition and aesthetic sense to see what fits together and what to study further.” What do you think?

The Merits of “Meritocracy”

Source: The Hedgehog Review, Summer 2016

as historian Joseph F. Kett has shown, in a fascinating and subtle study of merit’s travails through three centuries of American history, there are at least two strikingly different ways in which merit has been understood in that history.4 The founding generation itself thought in terms of what Kett calls “essential merit,” by which he means merit that rests on specific and visible achievements by an individual that were thought, in turn, to reflect that individual’s estimable character, quite apart from his social “rank.” “Merit” was that quality in the person that propelled the achievements, his “essential character.” Those who did the achieving were known as “men of merit,”

over time, a different way of understanding merit began to emerge, an ideal Kett calls “institutional merit.” Rather than focusing on questions of character, this new form of merit concerned itself with the acquisition of specialized knowledge, the kind that is susceptible of being taught in schools, tested in written examinations, and certified by expert-staffed credentialing bodies.

We would do well to leave room for the Lincolns among us—especially if they are as raw and uncredentialed as the man who would become our sixteenth president was. Think of his great speech at the dedication of the cemetery in Gettysburg in November 1863.

As many know, there were two notable speeches that day. The first, and the longest and most learned and most florid, was given by the supremely well-pedigreed Edward Everett, former president of Harvard—and the first American to receive a German PhD. But it was the self-educated frontiersman president who gave the speech whose accents ring down through the ages. Perhaps there is a pattern here to learn from.

Pokemon Go in South Korea

Source: The Verge, Jul 2016

Pokémon Go isn’t available in South Korea because the game uses data from Google Maps, which is restricted by the South Korean government due to security concerns. South Korea is still technically at war with North Korea, and the government has stated that it could release its map data only if Google deletes information on key security locations like military facilities. It’s unclear whether the game will ever be made available, but in the meanwhile, South Korean gamers have found a loophole.

The rhombus-shaped cells below show the areas that Niantic has labeled in its mapping system as restricted areas, and Sokcho just barely makes it out:

Since word got out that it was possible to play Pokémon Go in Sokcho, the town has been bombarded with tourists looking to play. Bus tickets from Seoul to Sokcho are sold out, according to the Associated Press, and tour packages including shuttle buses and hotel reservations have been popping up on deal websites:

The Mayor even caught a Machop (Fun fact: its Korean name translates to ‘Muscle Monster’) in this adorable Facebook Live interview with The Huffington Post in Korea.