The biggest mystery in mathematics: Shinichi Mochizuki and the impenetrable proof

The biggest mystery in mathematics: Shinichi Mochizuki and the impenetrable proof

"Sometime on the morning of 30 August 2012, Shinichi Mochizuki quietly posted four papers on his website. The papers were huge — more than 500 pages in all — packed densely with symbols, and the culmination of more than a decade of solitary work. They also had the potential to be an academic bombshell. In them, Mochizuki claimed to have solved the abc conjecture, a 27-year-old problem in number theory that no other mathematician had even come close to solving. If his proof was correct, it would be one of the most astounding achievements of mathematics this century and would completely revolutionize the study of equations with whole numbers."

The abc conjecture refers to numerical expressions of the type a + b = c. The statement, which comes in several slightly different versions, concerns the prime numbers that divide each of the quantities a, b and c. Every whole number, or integer, can be expressed in an essentially unique way as a product of prime numbers — those that cannot be further factored out into smaller whole numbers: for example, 15 = 3 × 5 or 84 = 2 × 2 × 3 × 7. In principle, the prime factors of a and b have no connection to those of their sum, c. But the abc conjecture links them together. It presumes, roughly, that if a lot of small primes divide a and b then only a few, large ones divide c.

But so far, the few who have understood the work have struggled to explain it to anyone else. “Everybody who I'm aware of who's come close to this stuff is quite reasonable, but afterwards they become incapable of communicating it,” says one mathematician who did not want his name to be mentioned. The situation, he says, reminds him of the Monty Python skit about a writer who jots down the world's funniest joke. Anyone who reads it dies from laughing and can never relate it to anyone else.

Reminded me first of Snow Crash.

The Singular Mind of Terry Tao

The NY Times wrote about The Singular Mind of Terry Tao "A prodigy grows up to become one of the greatest mathematicians in the world."

Longstanding problem put to rest

MIT News reports a Longstanding problem put to rest.

"The basic algorithm for determining how much two sequences of symbols have in common — the ‘edit distance’ between them — is now more than 40 years old. And for more than 40 years, computer science researchers have been trying to improve upon it, without much success. At the ACM Symposium on Theory of Computing (STOC) next week, MIT researchers will report that, in all likelihood, that’s because the algorithm is as good as it gets. If a widely held assumption about computational complexity is correct, then the problem of measuring the difference between two genomes — or texts, or speech samples, or anything else that can be represented as a string of symbols — can’t be solved more efficiently."

"Theoretical computer science is particularly concerned with a class of problems known as NP-complete. Most researchers believe that NP-complete problems take exponential time to solve, but no one’s been able to prove it. In their STOC paper, Indyk and his student Artūrs Bačkurs demonstrate that if it’s possible to solve the edit-distance problem in less-than-quadratic time, then it’s possible to solve an NP-complete problem in less-than-exponential time. Most researchers in the computational-complexity community will take that as strong evidence that no subquadratic solution to the edit-distance problem exists."

This viral math problem shows what American schools could learn from Singapore

Vox writes This viral math problem shows what American schools could learn from Singapore. 1. There's a viral math problem. 2. It's reasonably entertaining.

Albert and Bernard just became friends with Cheryl, and they want to know when her birthday is. Cheryl marks 10 possible dates: May 15, May 16, May 19, June 17, June 18, July 14, July 16, August 14, August 15, or August 17.

Then Cheryl tells Albert the month of her birthday, but not the day. She tells Bernard the day of her birthday, but not the month. Then she asked if they can figure it out.

Albert: I don't know when Cheryl's birthday is, but I know Bernard doesn't know either.

Bernard: At first I didn't know when Cheryl's birthday is, but now I know.

Albert: If you know, then I know too!

When is Cheryl's birthday?

P vs. NP and the Computational Complexity Zoo

Here's a nice 10 min video explaining a famous still unsolved math problem that every computer science major learns about. It even appeared in an episode of Elementary. Most of it should be understandable to everyone.

Stanford's Maryam Mirzakhani wins Fields Medal

Stanford's Maryam Mirzakhani wins Fields Medal "Maryam Mirzakhani, a professor of mathematics at Stanford, has been awarded the 2014 Fields Medal, the most prestigious honor in mathematics. Mirzakhani is the first woman to win the prize, widely regarded as the 'Nobel Prize of mathematics,' since it was established in 1936."

After 400 years, mathematicians find a new class of shapes

Ars Technica writes After 400 years, mathematicians find a new class of shapes "Platonic solids are generically termed equilateral convex polyhedra. In the millennia since Plato's time, only two other collections of equilateral convex polyhedra have been found: Archimedean solids (including the truncated icosahedron) and Kepler solids (including rhombic polyhedra). Nearly 400 years after the last class was described, mathematicians claim that they may have now identified a new, fourth class, which they call Goldberg polyhedra. In the process of making this discovery, they think they’ve demonstrated that an infinite number of these solids could exist."

1 + 2 + 3 + 4 + ⋯ = -1/12

This is making the rounds and I think this is the first time I've seen this. Apparently, the infinite sum 1 + 2 + 3 + 4 + ⋯ is -1/12. Really. And this value is useful in physics (well string theory and some other things). And I feel good because even the Bad Astronomer can't quite wrap his head around it.

This is the video going around...

After that one bothers you goto the followup page they posted with two more videos with some more details, Thanks for the messages. Each had a few bits that helped me.

There are some comments on Quora of people trying to help explain this non-obvious result. This one was useful.

Scientists Discover a Jewel at the Heart of Quantum Physics

I saw something about this before and thought I had blogged it but I can't find it now. Scientists Discover a Jewel at the Heart of Quantum Physics. "The revelation that particle interactions, the most basic events in nature, may be consequences of geometry significantly advances a decades-long effort to reformulate quantum field theory, the body of laws describing elementary particles and their interactions. Interactions that were previously calculated with mathematical formulas thousands of terms long can now be described by computing the volume of the corresponding jewel-like ‘amplituhedron,’ which yields an equivalent one-term expression."

Math with Bad Drawings

Math with Bad Drawings is pretty cute. "This blog is about the things I like. It’s also about the things I can’t do. I hope that the juxtaposition here – polished, thoughtful writing alongside art that my wife (charitably) likens to ‘the average 6th grader’ – captures the contradictory state of the teacher, of the mathematician – and, what the hell, of the human. We are all simultaneously experts and beginners, flaunting our talents while trying to cover our shortcomings the way an animal hides a wound. You could call this a ‘math blog,’ or a ‘teaching blog,’ but I would call it a blog about owning up to weakness and drawing strength from successes, however transient or trivial they may seem."

No Edge: The Shape of the Universe. (Part 1: Flat Models)

ZoggFromBetelgeuse explains some possible mathematical models of the shape of the universe in 10 mins. I learned a few things from it. So far parts 2 and 3 don't exist yet.

15 Uncanny Examples of the Golden Ratio in Nature

Every few years I see something like this io9 post, 15 Uncanny Examples of the Golden Ratio in Nature. I always forget about them and am always amazed at them.

The Value of Demos

When I learned a² + b² = c² I never thought of it like this:

Think you know all there is to know about paper snowflakes?

If you like math and have kids (or are a kid) and want to go crazy, then watch the video io9 points to, Think you know all there is to know about paper snowflakes? You pitiful, pitiful fool..

Meet the hexaflexagon. It’s about to blow your mind.

In this, her latest video, fast-thinking, faster-talking YouTube-maths-wizard Vi Hart presents us with the topologically fascinating hexaflexagon. First discovered in the 1930s by a daydreaming student named Arthur H. Stone, flexagons have attracted the curiosity of great scientists for decades, including Stone's friend and colleague Richard Feynman. Here, the ever-capable Hart introduces the folding, pinching, rotating, multifaceted geometric oddity with her signature brand of rapid-fire wit and exposition. She even shows you how to make your own. (via io9)

The Movie Math Quiz by Spiked Math

The Movie Math Quiz by Spiked Math is really really hard. Most of the films are reasonably well known but others are quite obscure. There's a lot of math I used to know.

John Nash’s Letter to the NSA

Turing's Invisible Hand wrote, John Nash’s Letter to the NSA.

"The National Security Agency (NSA) has recently declassified an amazing letter that John Nash sent to it in 1955.  It seems that around the year 1950 Nash tried to interest some US security organs (the NSA itself was only formally formed only in 1952) in an encryption machine of his design, but they did not seem to be interested."

"He then goes on to put forward an amazingly prescient analysis anticipating computational complexity theory as well as modern cryptography. In the letter, Nash takes a step beyond Shannon’s information-theoretic formalization of cryptography (without mentioning it) and proposes that security of encryption be based on computational hardness — this is exactly the transformation to modern cryptography made two decades later by the rest of the world (at least publicly…). He then goes on to explicitly focus on the distinction between polynomial time and exponential time computation, a crucial distinction which is the basis of computational complexity theory, but made only about a decade later by the rest of the world:"

"All in all, the letter anticipates computational complexity theory by a decade and modern cryptography by two decades. Not bad for someone whose “best known work is in game theory”. It is hard not to compare this letter to Goedel’s famous 1956 letter to von Neumann also anticipating complexity theory (but not cryptography). That both Nash and Goedel passed through Princeton may imply that these ideas were somehow “in the air” there."

How Our Primary Forecasts Work

FiveThirtyEight explains How Our Primary Forecasts Work

"One hugely mistaken assumption would be to look at the margin of error associated with the poll. FiveThirtyEight has a database consisting of thousands of primary and caucus polls dating back to the 1970s. Each poll contains numbers for several candidates, so there are a total of about 17,000 observations. How often does a candidate’s actual vote total fall within the theoretical margin of error?

The answer is, not very often. In theory, a candidate’s actual vote total should fall outside the margin of error only 5 percent of the time. In reality, the candidate’s vote total was outside the margin of error 65 percent of the time! Part of this is because the database includes some polls conducted months before the actual voting took place. But even if you restrict the analysis to polls conducted within the final week of the campaign, about 40 percent of the vote totals fell outside the margin of error — eight times more often than is supposed to happen if you could take the margin of error at face value."

Why Economic Models are Always Wrong

Scientific American has a nice little article Why Economic Models are Always Wrong. In other words, complex models are complex.

Re: Visual Multiplication and 48/2(9+3)

I saw this posted on BoingBoing by Cory Doctorow and thought it was cute: