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Luis Guzman
Senior Systems Analyst and Mathematics Graduate Student
Senior Systems Analyst and Mathematics Graduate Student


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Euclid's Element, Byrne's Edition. Completion of Byrne's work -- 6/13 of the volumes with colored figures.
KroneckerWallis already published Newton's Principa minimal design style --
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Easy as ABC? Not quite!

A brilliant mathematician named Shinichi Mochizuki claims to have proved the famous "abc conjecture" in number theory. That's great! There's just one problem: his proof is about 500 pages long, and almost nobody understands it, so mathematicians still can't tell if it's correct.

Luckily another mathematician named Go Yamashita has just written a summary of the proof. That's great! There's just one problem: it's 294 pages long, and it seems very hard to understand.

I'm no expert on number theory, so my opinion doesn't really matter. What's hard for me to understand may be easy for an expert!

But the most disturbing feature to me is that this new paper contains many theorems whose statements are over a page long... with the proof being just "Follows from the definitions".

Of course, every true theorem follows from the definitions. But the proof usually says how.

It's common to omit detailed proofs when one is summarizing someone else's work. But even a sketchy argument would help us understand what's going on.

This is part of a strange pattern surrounding Mochizuki's work. There was a conference in Oxford in 2015 aimed at helping expert number theorists understand it. Many of them found it frustrating. Brian Conrad wrote:

I don’t understand what caused the communication barrier that made it so difficult to answer questions in the final 2 days in a more illuminating manner. Certainly many of us had not read much in the papers before the meeting, but this does not explain the communication difficulties. Every time I would finally understand (as happened several times during the week) the intent of certain analogies or vague phrases that had previously mystified me (e.g., “dismantling scheme theory”), I still couldn’t see why those analogies and vague phrases were considered to be illuminating as written without being supplemented by more elaboration on the relevance to the context of the mathematical work.

At multiple times during the workshop we were shown lists of how many hours were invested by those who have already learned the theory and for how long person A has lectured on it to persons B and C. Such information shows admirable devotion and effort by those involved, but it is irrelevant to the evaluation and learning of mathematics. All of the arithmetic geometry experts in the audience have devoted countless hours to the study of difficult mathematical subjects, and I do not believe that any of us were ever guided or inspired by knowledge of hour-counts such as that. Nobody is convinced of the correctness of a proof by knowing how many hours have been devoted to explaining it to others; they are convinced by the force of ideas, not by the passage of time.

It's all very strange. Maybe Mochizuki is just a lot smarter than than us, and we're like dogs trying to learn calculus. Experts say he did a lot of brilliant work before his proof of the abc conjecture, so this is possible.

But, speaking as one dog to another: I can at least tell you what the abc conjecture says. It's about this equation:

a + b = c

Looks simple, right? Here a, b and c are positive integers that are relatively prime: they have no common factors except 1. If we let d be the product of the distinct prime factors of abc, the conjecture says that d is usually not much smaller than c.

More precisely, it says that if p > 1, there are only finitely many choices of relatively prime a,b,c with a + b = c and

d^p < c

It looks obscure when you first see it. It's famous because it has tons of consequences! It implies the Fermat–Catalan conjecture, the Thue–Siegel–Roth theorem, the Mordell conjecture, Vojta's conjecture (in dimension 1), the Erdős–Woods conjecture (except perhaps for a finitely many counterexamples)... blah blah blah... etcetera etcetera.

Let me just tell you the Fermat–Catalan conjecture, to give you a taste of this stuff. In fact I'll just tell you one special case of that conjecture: there are at most finitely many solutions of

x^3 + y^4 = z^7

where x,y,z are relatively prime positive integers. The numbers 3,4,7 aren't very special - they could be lots of other things. But the Fermat–Catalan conjecture has some fine print in it that rules out certain choices of these exponents.

It's a long way from here to the very first paragraph in the summary at the start of Yamashita's paper:

By combining a relative anabelian result (relative Grothendieck Conjecture over sub-p-adic felds (Theorem B.1)) and "hidden endomorphism" diagram (EllCusp) (resp. "hidden endomorphism" diagram (BelyiCusp)), we show absolute anabelian results: the elliptic cuspidalisation (Theorem 3.7) (resp. Belyi cuspidalisation (Theorem 3.8)). By using Belyi cuspidalisations, we obtain an absolute mono-anabelian reconstruction of the NF-portion of the base field and the function field (resp. the base field) of hyperbolic curves of strictly Belyi type over sub-p-adic fields (Theorem 3.17) (resp. over mixed characteristic local fields (Corollary 3.19)). This gives us the philosophy of arithmetical holomorphicity and mono-analyticity (Section 3.5), and the theory of Kummer isomorphism from Frobenius-like objects to etale-like objects (cf. Remark 3.19.2).

And it's a long way from this – which still sounds sorta like stuff I've heard
mathematicians say – to the scary theorems that crawl out of their caves around page 200!

Check out this paper and see what I mean:

You can read Brian Conrad's story of the Oxford conference here:

You can learn more about the abc conjecture here:

And you can learn more about Mochizuki here:

He is the leader of and the main contributor to one of major parts of modern number theory: anabelian geometry. His contributions include his famous solution of the Grothendieck conjecture in anabelian geometry about hyperbolic curves over number fields. He initiated and developed several other fundamental developments: absolute anabelian geometry, mono-anabelian geometry, and combinatorial anabelian geometry. Among other theories, Mochizuki introduced and developed Hodge–Arakelov theory, p-adic Teichmüller theory, the theory of frobenioids, and the etale theta-function theory.
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Some deep number theory lurking in an innocent-looking question: find a solution in positive integers to x/(y+z) + y/(x+z) + z/(x+y) = 4

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It's tragic that her life has ended at such a young age. I can't imagine how it will be for her young daughter Anahita and husband Jan.

My first memory of her was at a conference at Brown on Teichmuller theory. She stood in the back of the lecture hall due to pain from a back injury. After many talks, one would hear a voice pipe up from behind with insightful questions. During her talk, she spoke very quickly, as if she had a lot of information to communicate (which she did). I was impressed with the elegance of her viewpoint, but usually got lost partway through when she got to topics far from my area of knowledge.

In the fall of 2015, we convinced her to come to the Institute of Advanced Study in Princeton for a couple of months. Our daughters played together at a couple of the events and I got to meet her husband. She had the adjacent office to mine, and spent much of the time with Alex Eskin hashing through the 70 pages of referee comments on their 200+ page preprint. We chatted about math a few times when she asked me some questions about negatively curved surfaces. I quickly realized she knew way more than me about the problems, so I wasn't much help.

I'm saddened not just for her family, but also that she was cut short at the peak of her powers. We had so much more to learn from her.
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E8 from an icosahedral 'Big Bang'

My latest quest is to find a really simple, clear way to get E8 from the icosahedron. These are two of my favorite things, and they're connected.

The icosahedron is a Platonic solid with 120 = 1 × 2 × 3 × 4 × 5 symmetries. Just for fun, the picture here shows a 'stellated' icosahedron with sharper points. But it has all the same symmetries, and that's all that matters to me now.

E8 is an 8-dimensional lattice: a periodic pattern of points in 8 dimensions. This pattern gives the densest way to pack spheres in 8 dimensions: center a sphere at each lattice point, and make them big enough to just touch each other. Each sphere touches 240 others. That's the maximum possible in 8 dimensions. And in fact, if you pack spheres in 8 dimensions and get each to touch 240 others, you've got E8. This pattern shows up all over math, in cool and mysterious ways.

E8 has two little brothers. If you take a well-chosen slice of E8 you get a lattice called E7. This gives the densest known way to pack spheres in 7 dimensions. Similarly, if you take the right slice of E7 you get a lattice called E6, which gives the densest known way to pack spheres in 6 dimensions.

The McKay correspondence is a way to get E6, E7 and E8 from the tetrahedron, the octahedron and the icosahedron! This is one of nature's true marvels. It's yet more evidence that

In mathematics, everything sufficiently beautiful is connected.

There are actually several versions of the McKay correspondence. I'm interested in one called the 'geometric' McKay correspondence. Experts already understand it, but I want to bring it down to earth a bit... and I want to go for the jugular and focus on the icosahedron and E8.

My plan is to look at the space of all ways you can place an icosahedron of any size centered at the origin in 3d space. This space is 4 dimensional, since it takes 3 numbers to say how the icosahedron is rotated, and 1 more to say its size. And this 4-dimensional space has a singularity where the icosahedron shrinks down to zero size!

It reminds me ever so slightly of the Big Bang, where we have a 4-dimensional spacetime with a singularity where the universe shrinks down to zero size (roughly speaking). But this is just a cute analogy, the sort science journalists use to attract and confuse readers. The lazy readers only look at the headline, and come away with weird ideas. Don't be one of them.

The serious business here is seeing how E8 is lurking in the space of all possible icosahedra centered at the origin. Where is it?

It's sitting right at the singularity!

How? How is it sitting there, you ask?

I could tell you, but then I'd have to...

On second thought, I'll tell you here:

However, you'll need to know some math to follow this. The basic idea is that if you 'smooth out' or 'resolve' the singularity, it gets replaced by 8 spheres that intersect in a pattern governed by E8.

This is just the first part of a series, since there's a lot I still need to figure out! I want to see very concretely how these 8 spheres show up. I'm hoping some math friends of mine will help me. With luck, if we figure enough out, I can write a more polished article about it.

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Just ask Cleo

My real name is Cleo, I'm female. I have a medical condition that makes it very difficult for me to engage in conversations, or post long answers, sorry for that. I like math and do my best to be useful at this site, although I realize my answers might be not useful for everyone.

There's a website called Math StackExchange where people ask and answer questions. When hard integrals come up, Cleo often does them - with no explanation! She has a lot of fans now.

The integral here is a good example. When you replace ln³(1+x) by ln²(1+x) or just ln(1+x), the answers were already known. The answers involve the third Riemann zeta value:

ζ(3) = 1/1³ + 1/2³ + 1/3³ + 1/4³ + ...

They also involve the fourth polylogarithm function:

Li₄(x) = x + x²/2⁴ + x³/3⁴ + ...

Cleo found that the integral including ln³(1+x) can be done in a similar way - but it's much more complicated. She didn't explain her answer... but someone checked it with a computer and showed it was right to 1000 decimal places. Then someone gave a proof.

The number

ζ(3) = 1.202056903159594285399738161511449990764986292...

is famous because it was proved to be irrational only after a lot of struggle. Apéry found a proof in 1979. Even now, nobody is sure that the similar numbers ζ(5), ζ(7), ζ(9)... are irrational, though most of us believe it. The numbers ζ(2), ζ(4), ζ(6)... are much easier to handle. Euler figured out formulas for them involving powers of pi, and they're all irrational.

But here's a wonderful bit of progress: in 2001, Wadim Zudilin proved that at least one of the numbers ζ(5), ζ(7), ζ(9), and ζ(11) must be irrational. Sometimes we can only snatch tiny crumbs of knowledge from the math gods, but they're still precious.

For Cleo's posts, go here:

For more on ζ(3), go here:'s_constant

This number shows up in some physics problems, like computing the magnetic field produced by an electron! And that's just the tip of an iceberg: there are deep connections between Feynman diagrams, the numbers ζ(n), and mysterious mathematical entities glimpsed by Grothendieck, called 'motives'. Very roughly, a motive is what's left of a space if all you care about are the results of integrals over surfaces in this space.

The world record for computing digits of ζ(3) is currently held by Dipanjan Nag: in 2015 he computed 400,000,000,000 digits. But here's something cooler: David Broadhurst, who works on Feynman diagrams and numbers like ζ(n), has shown that there's a linear-time algorithm to compute the nth binary digit of ζ(3):

• David Broadhurst, Polylogarithmic ladders, hypergeometric series and the ten millionth digits of ζ(3) and ζ(5), available at

He exploits how Riemann zeta values ζ(n) are connected to polylogarithms... it's easy to see that

Liₙ(1) = ζ(n)

but at a deeper level this connection involves motives. For more on polylogarithms, go here:

Thanks to +David Roberts for pointing out Cleo's posts on Math StackExchange!

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A great graph theorist
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