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

Luis Guzman's posts

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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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Collatz Conjecture in Color:

Image courtesy of Josh Liu. Here is the source code if you want to generate the SVGs yourself:

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