Math(s)! Currently: Summer of Math Exposition

There certainly haven't been any serious claims on proving all of BSD (I'm not even aware of any cranky ones, but I'm sure if you looked hard enough you could find them). But I think that's by nature of the problem: solving BSD would require a completely new construction (or a completely unexpected classification) as the first step of the proof, and no one has any good candidates already out there. If you prove BSD then you've shown that any time L(E,s) has a zero of order 3 at s=1, then you can find three linearly independent elements in E(Q). Barring some incredibly bizarre proof strategy this is going to require either (a) coming up with a systematic way of constructing points on a broad infinite class of elliptic curves or (b) classifying all of the curves you need to deal with in an explicit enough way to get these points by ad-hoc methods. And no one has any good ideas of how to get started doing either of those things! (On the other hand, if L(E,s) has a zero of order 1 then there is a construction of something called a Heegner point that gives an element of infinite order in E(Q), and that idea has been hugely instrumental in all of the progress that's been made in the cases of rank 1 elliptic curves).

This is in contrast with the case of P=NP, where you already have the object you need to look at: any of the NP-complete problems that we have lying around. You don't need to go and build a new problem from scratch and then try to prove it's in NP and not P, you instead work with something you're familiar with. So that seems more likely to lead to serious attempts to analyze those problems and perhaps to get something that seems like a proof but actually has a flaw you've overlooked. (Just to be clear, I'm not saying that P=NP is an easier problem than BSD, just that the hurdles involved aren't so blatant right from the beginning). My vague understanding of the Hodge conjecture is that it's similar to BSD in that it's essentially asking to construct things that no one has any idea how to build systematically (and I don't know of any claims to have proven that either). The Riemann hypothesis is more along the lines of P=NP in that you're trying to prove something can't happen with some mathematical object you already have sitting in front of you, and it's a lot easier to get drawn into a flawed proof attempt without realizing it.

 

Why? I mean, I get that the Standard Model is messy, but like, wouldn't proving Yang-Mills to be true actually clean it up a bit?

EDIT:

@Emu something something physicists trying to solve rigorous mathematical problems by playing with non-hermitian operators
How did that even get resolved? Did the comment-on-a-comment-on-a-comment-on-a-comment-on-a-comment-on-a-comment finally refute it or...?

 

There are a few aspects to judge a physical theory on.
There's mathematical rigor: the theory explicitly presumes some axioms and derives its phenomenological claims from those axioms.
There's mathematical elegance: the theory presumes very few axioms.
There's ontological elegance: the theory presumes very few fundamental objects.
There's empirical observability: the theory produces some phenomenological claims along with means to empirically test those claims.
There's empirical accuracy: the theory's observable aspects match well with observed measurements.

The Standard Model does pretty well on the empirical side. QED in particular has bragging rights from its prediction of the electron's magnetic moment. Sure, there's the muon g-2 anomaly, but whatever, right?

However, there are a bunch of issues with the rigor and elegance sides.
On the rigor side there are a bunch of holes in the logic, assumptions that are never made explicit, ideas that are never concretely defined, claims that are never justified. This is where Yang-Mills comes in, to fill in the logical hole in non-Abelian gauge theory that currently says "we haven't shown that nontrivial non-Abelian gauge theories satisfying some basic properties actually exist". I don't mind this particular issue all that much, because there's a lot of interesting mathematics coming out of it, which in my book makes it a good problem. Not a good problem for a theory to have, but a good problem for the field of mathematical physics to have. Yeah, integration-over-histories and renormalization and all those flyaway infinities are worrisome, but that's a problem with field theory in general.
However, the Standard Model is also ugly. 19 arbitrary parameters. Kludgy mechanisms for explaining mass and mass discrepancies. It's a bunch of disparate pieces stuck together like a video game that was passed through three different studios each working with a different engine. It's great that it manages to unify the three fundamental forces into a single framework, but when it comes to explaining why those fundamental forces appear different despite the supposed unification, it resorts to arbitrarity and contrivance.

So even if they patch the rigor holes, even if they make a clean, mathematically solid path between explicit axioms and phenomenology, it's still ugly, and that's why I hate it.

 

The paper Eric's talking about is this one, which I'd noticed when it showed up in the number theory section of the arxiv a while back because of the title Comment on 'Comment on "Hamiltonian for the zeros of the Riemann zeta function" ' . Just glancing at things there's no more arxiv postings by any of the people involved, and also not any serious consideration of the idea by mathematicians.

So the basic idea here is a well-known way of approaching the Riemann hypothesis, actually. You're trying to show all of the zeros of the Riemann zeta function are on a certain line, and the idea is to translate the domain so the line you're going for is the real axis, then try to construct an operator on a Hilbert space such that (a) the eigenvalues are exactly the (translated) zeros of the zeta function, and (b) the eigenvalues are all real. Of course no one's done (a) and (b) together since that would be a proof of RH! And I don't know of any proposed constructions of such an operator that are considered promising - obviously (b) would be trivial if the operator was Hermitian, but when people have constructed reasonable operators from spectral theory satisfying (a) they're non-Hermitian with no way of getting a handle on the eigenvalues.

In this case, my vague understanding is that the physicists who wrote the original paper proposed an operator satisfying (a), and said that it's plausible that it fits into some sort of theoretical physics framework that implies (b). And then they proceeded to publish it in a fairly prominent physics journal, attracting some attention in the science media, but ultimately just leading the mathematicians looking at it to rub their temples and say "there's nothing that seems new or interesting here, and also trying to rigorously define the things proposed in the paper leads to a huge mess." Hence the first 'comment on' paper, which I guess the second 'comment on' paper didn't actually address in a way that made any of the mathematicians reconsider.

 

Ah, k. Yeah, I really hate that ugliness too. Makes sense.

 

So a while ago people were wondering why we care about efficient sphere packing in high dimensions. And the answer is: so we can send messages to space.

[Cue theme music]

Today's topic: Error Correcting Codes!

Suppose you want to tell your friend something but you're in a crowded room and everyone else is talking. You say some words but there's a lot of noise, and your friend only hears maybe one word in four. That's probably okay if the message is something simple like "fuck you" but not if it's more complicated. Too much of your message is being lost in transmission.

So what can you do?
Well, the usual way this works is that your friend says "What?" and you repeat yourself. And this process continues until your friend figures out what you're trying to say or you get sick of not being heard.
We call this a repetition code. It's simple and straightforward, but not terribly efficient.

So let's do this a bit more technically. Instead of shouting words in a noisy room, let's abstract. You're trying to transmit a sequence of 0s and 1s, and you know that whatever you're using to transmit this sequence over has noise, i.e. will sometimes replace a 0 with a 1 or vice-versa. How can you deal with this issue?

Well, we can do a repetition code. Each time we have a 0 in our sequence, we transmit 3 0s, and each time we have a 1 in our sequence, we transmit a 1. So the sequence '010' becomes '000111000'. But suppose there's a bit of noise and instead the receiver gets '010110001' instead.
Now if our receiver gets a sequence of 0s and 1s, they figure out what the original message was by looking at triplets and taking a majority vote. So if they get the triplet '010', they figure the original was probably a 0, and if they get '110', they figure the original was probably a 1.
So our receiver takes triples, takes the majority vote for each triple, and successfully figures out that our original sequence was '010'.
Great!
Except not terribly efficient. We have to send 3 bits of message for each bit of original sequence, and we can only handle one bit of error per every 3 bits of message. If the original sequence as '0', and '000' got transmitted, but there were two errors and the receiver got '110', then the receiver thinks that the original message was '1'. Not great!

Claude Shannon, the guy who basically invented the entire modern notion of information, showed that we can definitely do better than this.

So let's be a bit more mathematical about this.
We want to consider an error correcting code to be a map from k-bit sequences to n-bit sequences for some numbers n and k; we'll call the k-bit sequences words, which is the message we want the receiver to get, and the n-bit sequences we'll call codewords, which is what are sent out. In other words, we take k-bit long words, map each word to a codeword, and send the codeword.
The receiver gets a sequence of n bits and tries to figure out which k-bit word probably yielded that n bit sequence, assuming some noise.
Now if n is less than k, this obviously won't work because then we have fewer possible codewords than we have words, so information is getting irretrievably lost even before anything is transmitted. If n is equal to k, then there's no gain from using this code because there's nothing protecting against noise; every possible n-bit sequence received looks like it came from a k-bit word, so the receiver has no way of telling if anything went wrong during transmission.
So we assume n > k, and that not all n-bit sequences are codewords. Like the repetition code, this redundancy gives us some breathing room to deal with noise.

In 1948, along with defining what information is, Shannon proved that for a fixed probability of any given bit flipping during transmission, the maximum number of distinct codewords of n-bits that you can have before you expect the receiver to screw up trying to decode them is exponential in n, and as a proportion of all n-bit sequences gets larger as n gets larger.
Of course, that doesn't necessarily help you figure out which codewords to use, or how the receiver is supposed to decode them.

If we assume that there isn't too much noise, then given a received n-bit sequence, that sequence probably started life out as which ever n-bit codeword is closest to the received sequence. What does it mean for two sequences to be close? We use the same idea as in the Gray codes post: total number of differences. So '110' and '101' are a distance 2 apart, while '110' and '111' are a distance 1 apart, and so on. This is called the Hamming distance.
The farther apart our codewords are, the better our code will be because then the receiver can distinguish between n-bit codewords more easily. If we have two codewords that are only distance 1 apart, then a single error will make one look like the other, which means the receiver will be confused. However, if the minimum distance between codewords is 2, then a single error in a codeword can be detected, because the resulting received word won't be a codeword. Moreover, if the minimum distance is 3, then a single error can be corrected, because now any n-bit sequence is either a codeword or is exactly distance 1 away from a single codeword, and so we can correct to that particular codeword.
The general rule is that if you want to be able to detect up to t errors, your minimum distance between codewords must be at least 2t, and if you want to be able to correct up to t errors, your minimum distance between codewords must be at least 2t + 1.
So when making error correcting codes, a natural question to ask is "can we pick n-bit codewords so that each pair of codewords is at least distance 2t (or 2t+1) apart"?
If you think of your set of n-bit codewords as an n-dimensional space, a dimension for each bit, then the question above becomes how to pick points in this space to be codewords. We can consider the points within distance t of our codewords, these being the n-bit received sequences that will be corrected to that particular codeword. Usually we call things a fixed distance away from a single point a "sphere", and then the question of how to pick codewords becomes a sphere-packing problem.

Okay, I'm being a bit deceptive here. Yes, it's a "sphere-packing problem in n-dimensional space", only these dimensions are just bits, so that there's only two possible values in any direction, and these "spheres" are spheres according to the Hamming distance function, which doesn't yield things that look all that much like spheres in Euclidean space.
But it turns out that there are ways to translate the Euclidean sphere-packing problem in n-real-dimensional space into the Hamming sphere-packing problem in n-bit-dimensional space. In particular, the E8 lattice gives the best solution to the Euclidean sphere-packing problem in 8-real-dimensional space, and translates into the Extended Hamming [8,4,4] code, which takes 4-bit words, turns them into 8-bit codewords, and has codewords at least a distance of 4 apart (hence can detect 2 errors per codeword and can correct 1).
Similarly, the Leech Lattice, which solves the Euclidean sphere-packing problem in 24-real-dimensional space, becomes the Extended Golay [24, 12, 8] code, which takes 12-bit words, turns them into 24-bit code words, and has minimum distance 8 (hence can detect 4 errors per codeword and correct 3). This code was used for transmissions to and from the Voyager 1 and 2 spacecrafts, since they had a lot of data but space is noisy (in the radiation sense, not in the audio sense).

The field of error correcting codes actually has a ton of really cool mathematics in it. A lot of codes these days are built on ideas from abstract algebra, in particular finite fields and the theory of polynomials over said finite fields.
The main issue in designing an error correcting code is actually computational efficiency. It's not necessarily difficult to create a code that has a decent minimum distance between codewords without too much redundancy, but in particular computing which codeword is closest to a given n-bit sequence is difficult for most codes with said decent minimum distances. It's only relatively recently (in the 90s) that we've started to even approach the Shannon limit (i.e. for a given amount of noise, we approach the number of codewords given by the function that Shannon found).

Anyway, that's why the sphere-packing problem in 8 and 24 dimensions was actually one of the most widely applied pieces of mathematics, although the corresponding Hamming and Golay codes have been replaced with more efficient codes.

 

Maunderings on Mochizuki

I am not a number theorist, unlike some people here, so I have no idea whether the IUT situation is actually getting any clearer or not. My general impression is "not significantly."
Go Yamashita recently released a 290-page paper attempting to boil down Mochizuki's 500+ page paper, with the unfortunate effect of several-page-long statements of theorems being followed by "Proof: follows from the definitions".
At least, I'm assuming this is unfortunate.

There are, to my understanding, two issues with Mochizuki's proof: we (as in, people who aren't Mochizuki or apparently Yamashita) don't know if the proof works, and we don't understand the proof, regardless of its correctness.

In theory, we could dump the proof into Coq or Agda or some automated proof-verifier program, hit "run", and wait for it to spit out a "correct" or not. In theory this would not require understanding any more than how to make the various definitions and sub-proofs precise enough for a computer to parse, which itself ought to be somewhat mechanical of an endeavor, provided that the source material is detailed enough.
In practice, this is challenging because the source material relies on a lot of other work, both the author's and others', that has not be made machine-readable, and so a lot of effort would have to go into preparing this background material even before approaching the new material. Automated proof-verification is still in its infancy, and people are still building the libraries that in a more mature system could be easily referenced and imported. We currently cannot just write "include algebraic_geometry.lib". Someday we will be able to, but not yet.
So verifying the proof via automation would require building a lot of framework. That framework could be used for verifying other proofs, but it needs to be built.

Conversely, we have the question of making the proof human-readable. Mochizuki introduced a lot of new language in his paper, which is still being examined to see if it can be translated into language compatible with the standard mathematical tongue. He came up with a bunch of new ideas which are still being examined to see if they make any sense, and if they can be used to illuminate rather than obscure.
A while back I discussed the generalized Stokes' theorem, and why it was considered a mathematically well-developed theorem: the objects were natural objects to consider, and were defined and described in such a way that the theorem was obviously true and the proof would clearly work even with the details removed. We are not there yet with Mochizuki. Are the objects natural objects to consider? Should we expect them to act the way Mochizuki claims they do, even without verifying all the details? Do we understand enough to have beliefs?

And what if Mochizuki's proof doesn't work? What if there is a fatal flaw?
Again, nonexpert here, but I get the impression that, even if there is a fatal flaw, there should be salvageable ideas in the proof. Mochizuki has done good work in the past. Sure, his work on the abc conjecture is mostly impenetrable, but there is definitely something there, something solid and usable, if not for proving the abc conjecture.

 

I am a number theorist but don't have any inside info on what people think about Mochizuki's proof, so I can't really add much to what you've said. There's been some progress with conferences/workshops getting the initial layers of his theory across to more experts in arithmetic geometry. but it still seems to be largely a stalemate between "show me some interesting new idea from this theory that I can understand without devoting a massive amount of time to it" versus "you need to carefully understand all of the foundations that are set up before getting to the consequences".

 

"Almost"

Mathematics has a precise definition of "almost", or at least a collection of related terms which involve the word "almost".

Consider the following: if you wanted to run 50 kilometers and managed to make it 49 kilometers, you might say that you "almost" ran 50 kilometers. But you didn't make it all the way, and while 98% is often good enough, sometimes it isn't.
If you wanted to drive 500 kilometers and managed to make it 499 kilometers, you might say that you "almost" drove 500 kilometers. But you didn't make it all the way, and while 99.8% is usually good enough, sometimes it isn't.
If you wanted to fly 5000 kilometers and make it 4999 kilometers, etc etc.
In mathematics, the notion of "almost" is attached to things that aren't quite all the way, but are close enough for any notion of close enough that isn't all the way, regardless how stringent.

Consider the phrase "Almost all natural numbers are bigger than 1". This is true, but it is also a mathematically precise statement. The collection of natural numbers that are not bigger than 1 is finite (very much so) while the collection of natural numbers that are bigger than 1 is infinite. But trying to express this as a percentage is problematic because while it's misleading to say that 100% of natural numbers are bigger than 1, it's also misleading to say that X% of natural numbers are bigger than 1 for any real number X less than 100.
So instead we have the perfectly serviceable "Almost all".

Consider the phrase "If you flip a fair coin an infinite number of times, you will almost surely get at least one head and at least one tail". It is mathematically possible to flip an infinite string of all heads, or an infinite string of all tails, but the probability for either of those outcomes is 0 (not impossible, but smaller than any finite non-zero number). So it's not the case that you surely will get at least one head and at least one tail, but it's almost sure.

It's not even that you're always missing out on a finite set; almost all real numbers are irrational, in that if you were to pick a real number at random* then the probability of picking a rational one is 0, despite there being an infinite number of rationals.

Nor is it even about different sizes of infinities. If you take the interval [0, 1] and consider the Cantor set, both the interval and the Cantor set have the same cardinality, but almost all points in the interval are not in the Cantor set.

There's a lot of fun you can have with "almost". For instance, there's a function called "the devil's staircase" or the Cantor staircase which is continuous, has value 0 at x = 0, value 1 at x = 1, and has derivative 0 almost everywhere. In other words, it's constant except at isolated points, but still manages to get from 0 to 1 while still being continuous at the non-constant points.


Anyway, I bring this up because there's an article in Quanta Magazine about how some mathematicians proved that two infinities are equal, but it's a garbage article and I suggest you don't read it because it encourages ignorance and intellectual negligence and as part of that encouragement of ignorance it doesn't bother to explain which infinities, instead choosing to make it seem like the Continuum Hypothesis had been proved true.


Anyway, the actual infinities in question are pretty interesting, so I'm going to attempt to explain them.
We start with the set of natural numbers, nice and familiar. Countable.
Then we consider infinite subsets of the natural numbers. Each subset is countable (I guess we're assuming the axiom of Choice or something). From now on, when I say "set" I mean a subset of the natural numbers.
We define a notion of "pseudo-inclusion": a set A is pseudo-included in set B if all but a finite number of elements of A are also in B. So, for instance, the set of all natural numbers is pseudo-included in the set of all natural numbers bigger than 1, and of course vice-versa. We'll just write it as <= because I'm lazy: A <= B if only a finite number of elements in A are not in B.

We say that a collection of subsets of the natural numbers has a pseudo-intersection if there is an infinite subset A such that A <= B for all sets B in the collection. Note that this is not necessarily the same as the actual intersection, since A doesn't actually have to be a subset of any of the sets in the collection.

Puzzle 1: Find a collection of subsets of the natural numbers that has a pseudo-intersection, but where the actual intersection of all of the elements of the collection is empty.

We can now ask about collections of subsets for which pseudo-inclusion makes up a linear order. In other words,
For A and B in the collection, either A <= B or B <= A
If A <= B and B <= A then A = B
If A <= B and B <= C then A <= C
We say that a collection is a tower if pseudo-inclusion make a linear order, but the collection does not have a pseudo-intersection.
So towers look at collections that have a funny ordering scheme where the "smaller" sets are almost contained in the "bigger" ones, and in which there is no "smallest" set that sits inside all of the "bigger" ones.
We say that t is the smallest possible cardinality for a tower.

Now consider a collection of infinite subsets of the natural numbers. We say such a collection has the finite intersection property if for any finite subcollection, the intersection of all of those sets in the subcollection is nonempty. In other words, if there is some element that is in every subset in the subcollection.
We say that a collection has the strong finite intersection property if for any finite subcollection, the intersection of all of the sets in the subcollection is infinite.
We say that p is the smallest possible cardinality for a collection with the strong finite intersection property and without a pseudo-intersection.

So the notion of "almost" is key to both t and p.

Puzzle 2: Convince yourself that every tower has the strong finite intersection property. Hence t is at least as large as p.

Hausdorff showed that both p and t are at least as big as alpha₁, the smallest uncountable infinity. So if the Continuum Hypothesis is true, then both p and t have to be the size of the real numbers. But without the Continuum Hypothesis, there could be a bunch of infinities between the cardinality of the natural numbers and the cardinality of the reals, and so it wasn't known if they had to be the same size.

Anyway, the result, by Maryanthe Malliarisa and Saharon Shelah, says that yes, p = t.

* Okay, this is a little wonky because you can't stick a uniform distribution on the reals in a nice way, so it's better to talk about measure theory, but I don't want to talk about measure theory.

 

I really like how people working on higher-dimensional algebra just have all sorts of -hedra to talk about coherence laws and such. I just really like how that whole cluster of higher-categorical mathematics really digs into the idea that diagrams and higher-dimensional analogues are mathematically precise enough to be studied and give results about higher categories.
Plus the names are really fun. "Permutohedra", "associahedra".

 

Coffins

In the 1970s and 80s, Soviet math departments used a special examination to exclude Jews from being able to enter. A set of problems, called "coffins", were devised based on three criteria:

1): The problems could be easily stated with only fairly elementary mathematics.
2): The solutions could be easily stated with only fairly elementary mathematics.
3): Finding said solutions required unmotivated leaps of faith, constructing things or doing computations that, at the outset, appear totally irrelevant to solving the problem at hand.

It was only after coming to the US that former Soviet mathematicians start discussing these problems in public. Now several articles have been written on them, and many have published solutions.
And these are devilish puzzles. Some of them are simply awful computations that look like they should be easier than they are; some just require completely out-of-the-box thinking. A few are straight-up wrong.

One could compare them to International Mathematics Olympiad problems, which are famed for their difficulty; the difference is that in the Olympiad, you're given two days to write out solutions six problems. In the case of coffins, it's an oral exam that continues until you fail. Criterion 3 usually guarantees eventual failure, and criteria 1 and 2 give plausible deniability to the examiners.

Anyway, if you're feeling masochistic, you might try some of the coffins, partially available here and here thanks to Tanya Khovanova. I think everything listed has solutions, and some of the solutions are really quite cute, but are generally "why would you even think to do that?"
Alternatively, if you're like me, you might suggest a few to your mathematically inclined friends. I'm not saying that you should, but you might.

 

Oral exam problem:

 

I'm not really sure that I'd say that has a solution that's easy to state.

 

it does have a solution that's easy to state

the solution is "no, you can't"

 

It's pretty much impossible to solve unless you 1) recognize it as a cubic 2) have access to e.g. Sage

 

Lattice rings!

Okay, so the number theorists in the audience are probably going to be a little irritated at me for sliding a bunch of stuff under the rug in this post, but that's a risk I'm willing to take. In particular, please ignore any issues with the scaling of the lattices.

So the integers. Nice, familiar*, the integers have some pretty basic algebraic operations available. You can add two integers, or subtract them, and you can multiply two integers, and in all of these cases you will get another integer. These are the ring operations, and we like them, even though you can't necessarily divide. The integers contain 0, and 1. And they're associative and commutative. Also they're discrete; there's a minimum distance between integers. The integers form a lattice.
And the integers are the best you're going to get out of subsets of the real numbers. There's no lattice of real numbers with the ring operations and 0 and 1 other than the integers. If you try to include less you are going to lose 1 or you're going to lose the ring operations; if you try to include more you're going to lose discreteness.

You have a bit more freedom in the complex numbers. For instance, you can look at numbers of the form a + bi, where a and b are integers. Numbers of the form a + bi are called Gaussian integers, and they're pretty great. You can check that these guys are closed under the ring operations, and of course 0 = 0 + 0i and 1 = 1 + 0i are Gaussian integers. Also, there's a minimum distance between Gaussian integers, so we get a lattice.
We can also scale our imaginary axis, for instance picking a positive integer M; then the elements of the form a + (b√M)i also form a lattice ring.

What other kinds of lattices do we get in 2 real dimensions? There's the triangular lattice. And is there a set of complex numbers that form a triangular lattice and are closed under the ring operations and contain 0 and 1? Yes! The Eisenstein integers are complex numbers of the form a + bw where a and b are integers and w is a complex cube root of 1; you can choose it to be -1/2 + (√3)i/2 if you want something concrete. And you can again check that the Eisenstein integers are closed under the ring operations. Hint: w² = -1 - w.
And again, you can check that these form a lattice.

We have lattices in 3 dimensions, but we don't have any division algebras, or even any really interesting algebras, so let's move upward.

The quaternions. We can make a ring a + bi + cj + dk, again with a, b, c and d integers, and i, j and k are all square roots of -1 that anticommute: ij = k = -ji and etc. We call these the Lipschitz integers, or Lipschitz quaternions. And again this is a lattice and is closed under the ring operations, although it's not commutative anymore. Still, a little bit boring, if you ask me. They form a nice hypercubical lattice, which is A₁⁴ in the ADE stuff I talked about earlier.
You can also do some scaling of your basis elements like in the complex case, although here while you can scale two of the imaginary basis elements as you please, you need to make sure that the third scaling is compatible with the product of the other two.

But let's do something a bit more complicated.

We can make a ring a + bi + cj + dk where either all of a, b, c and d are integers, or all of them are half-integers, by which I mean of the form n + 1/2 for n an integer. These are called the Hurwitz integers. They form a D₄ lattice, which you can think of as three copies of the A₁⁴ lattice, two of them offset from the original.
These are more interesting! In particular, like the regular integers, the Gaussian integers and the Eisenstein integers, and unlike the Lipschitz integers, there's a notion of "division-with-remainder" for the Hurwitz integers! Here we mean that given a and b, we can find q and r such that a = bq + r, and r is "smaller" than b for some appropriate notion of "smaller". For the Hurwitz integers, we can use the quaternion norm to define "smaller".

So let's move on to dimension 8. The octonions. Now we won't have things that are commutative or associative. But we can still look for lattices that are closed under the ring operations.

Let's take a moment to recall how to multiply in the octonions. We have eight basic elements, which I'll call e₀, e₁, ..., e₇, where e₀ is just a fancy way of writing 1. The rest of them square to -1.
I'm going to write down the following triples:
124, 235, 346, 457, 561, 672, 713
and we can figure out the multiplication from these triples. Suppose you have a triple pqr. Then we say that
e_pe_q = e_r = -e_qe_p
e_qe_r = e_p = -e_re_q
e_re_p = e_q = -e_pe_r
So let's look at lattices. There are a bunch of lattices in 8 dimensions, but only a few actually do what we want.
There's the obvious one, where we have a₀e₀+a₁e₁+...+a₇e₇, where the a_i are all integers. This is the Gravesian integers. The form a nice hypercubical lattice, or an A₁⁸ lattice.
There's the case where the a_i are all integers or all half-integers. These are the Kleinian integers. They're closed under the ring operations for the same reasons that the Hurwitz integers are, and analogously to the Hurwitz integers they form a D₈ lattice.
But we can do better. Pick a triple, pqr. Consider all octonions of the form (pm 1 pm e_p pm e_q pm e_r)/2, where pm means plus or minus, and all the octonions of the form (pm e_s pm e_t pm e_u pm e_v) where s, t, u and v are the numbers between 1 and 7 (inclusive) that aren't in the triple, and all the numbers you can make by adding and subtracting these particular octonions.
These are called the Double Hurwitz integers, and they're also closed under the ring operations for any of the triples. As a lattice, they're D₄².
But there's seven distinct sets of Double Hurwitz integers given our basis octonions, so obviously there's some way to piece them together to make a lattice that doesn't depend on picking a triple first, right?
So let's take all seven triples, and the corresponding sets of Double Hurwitz integers, and just take their union; take everything that's in at least one of the sets. These are called the Kirmse integers. Which is kind of a mean thing to do, because Kirmse, who first formulated this set, didn't realize at the time that they aren't closed under multiplication.
There is a fix! Pick a number n between 1 and 7 (inclusive). Then take all of the Kirmse integers, with the form a₀e₀ + a₁e₁ + ... + a₇e₇, and swap the coefficients a₀ and a_n. This does turn out to be closed under multiplication, and gives us the Cayley integers. Or at least, one of seven flavors of Cayley integer. And, as with all things 8-dimensional, they form an E₈ lattice.

*Actually kind of bizarre and terrifying; the job of a number theorist is to keep them at bay so that the rest of us can pretend that we understand them. This is subtly different from the job of the model theorist, who's solemn duty is to convince us that there's only one set of integers and that this set can be reasoned with by said number theorists; in truth this may be false; in truth this may be unknowable. It is said that God created the integers, but the god that created them may be more Eldritch than Abrahamic.

[EDIT] Mentioned some places where you can scale stuff, despite saying that I was going to brush all of that under the rug.

 

I want to write a post about something relating to Homotopy Type Theory in honor of Voevodsky's passing. While I'm neither a logician nor a homotopy theorist by any stretch, I did read the HoTT book when it first came out and found it fascinating. Everything is a type and everything is a space and everything is an infinity-groupoid. Metamathematics is just the mathematics of the type/space/infinity-groupoid Math. Two things are equivalent if they are isomorphic if they are connected by a path. Logical "statements" can have meaningful structure beyond just variations of True or False.
I was never really interested in logic or algebraic topology at all, but this stuff was super cool, a much richer, more dynamic framework for thinking about mathematics than set theory, at least in my opinion. While it might not have changed my mathematical career at all, it definitely changed how I think about mathematics as a study and as an object of study.
Unfortunately, the founder of HoTT, Vladimir Voevodsky, passed away a few days ago. I wish I could write more coherently about his work.

 

There are other lattices in the quaternions

 

Oh, right. I don't know if they have names, but yeah, you can do things like the Gaussian integers times the Eisensteins. I suppose that if I'm going to include the Double Hurwitz integers I might as well include these ones as well.

 

Yeah. One of them can be generated by the elements of a double cover of S_3 in much the same way the Hurwitz quaternions can from a double cover of A_4, for instance.