highergeometer

johncarlosbaez, 5 days ago (edited 4 days ago) to random

We're getting close to understanding the deep connection between the cross product in 3 dimensions (which they may have taught you in college) and the cross product in 7 dimensions (which they almost certainly did not). They are not separate things! It seems you can define the latter in terms of the former!

What this means - like how it's related to the rare earth elements called 'lanthanides' - remains obscure. But we're miles ahead from where we were half a month ago:

https://golem.ph.utexas.edu/category/2024/06/3d_rotations_and_the_7d_cross.html

highergeometer, 6 days ago to random

DOIs for Theory and Applications of Categories!!

Nearly 11 years ago I spoke with the then editor-in-chief of the open access journal Theory and Applications of Categories (TAC) about the option of registering Digital Object Identifiers (DOIs) for articles. This is a modern piece of publishing infrastructure without which a journal looks it is still living in the 1990s—which was when TAC was started, in the first wave of new, online journals that were designed to be low-frills and run by academics.

https://thehighergeometer.wordpress.com/2024/06/03/dois-for-theory-and-applications-of-categories/

highergeometer, 9 days ago to random

A colleague asked what I thought ((\frac10)^0) should be.

What are your thoughts?

I had an idea about something related to this, but I want to see others' comments first. I'll reply to this with a spoilered text with that idea.

#mtbos

highergeometer, 9 days ago to random

Steve Vckers speaking at the Topos Institute: the real line is not a set 😏

https://www.youtube.com/watch?v=L6LPEFteLts

johncarlosbaez, 11 days ago (edited 11 days ago) to random

The Kervaire Invariant Problem is a famous problem in topology that remains open in just one case: the case of 126 dimensions! See the Wikipedia article for an explanation.

Apparently this one remaining case has now been solved by Zhouli Xu, Weinan Lin and Guozhen Wang. Xu will reveal the answer tomorrow, Thursday May 30th, at the Princeton Algebraic Topology Seminar! I think a video will appear later online.

.....

Computing differentials in the Adams spectral sequence
May 30, 2024 - 01:00 - May 30, 2024 - 02:00
Zhouli Xu, University of California, San Diego

Online Talk

I will review classical methods computing differentials in the Adams spectral sequence, and then discuss some recent progress in joint work with Weinan Lin and Guozhen Wang. In particular, I will discuss the fate of h_6^2, resolving the Last Kervaire Invariant Problem in dimension 126.

https://en.wikipedia.org/wiki/Kervaire_invariant

johncarlosbaez, 15 days ago (edited 15 days ago) to random

Robert Recorde introduced the equal sign in 1557. He used parallel lines because "no two things can be more equal". And his equal sign was hilariously looooooong.

This is from @mjd's excellent blog article:

https://blog.plover.com/math/recorde.html

and I recommend following him here on Mastodon.

It's fun to fight your way through Recorde's text, with its old font and spellings. But if you give up, @mjd has transliterated it:

Howbeit, for easie alteration of equations. I will propounde a fewe exanples, bicause the extraction of their rootes, maie the more aptly bee wroughte. And to avoide the tediouse repetition of these woordes "is equalle to" I will sette as I doe often in woorke use, a pair of paralleles, or Gemowe lines of one lengthe, thus: =====, bicause noe 2 thynges, can be moare equalle.

The only real mystery here is "Gemowe", which means "identical" and comes from the same root as "Gemini": twins.

In the same book Robert Recorde introduced the mathematical term "zenzizenzizenzike", but I'm afraid for that you'll have to read @mjd's article!

highergeometer, 16 days ago to random

Hooley dooley.

This post is about a "found-in-the-wild" small Turing machine (3-states, 4-symbols), that halts --- after an Ackermann function-level number of steps:
[ 14 \uparrow\uparrow\uparrow\uparrow\uparrow\uparrow\uparrow\uparrow\uparrow\uparrow\uparrow\uparrow\uparrow\uparrow 14
]
and in fact we know exactly how many!

https://www.sligocki.com//2024/05/22/bb-3-4-a14.html

highergeometer, 16 days ago to random

Two thematically related papers

Felix Cherubini, Thierry Coquand, Matthias Hutzler, David Wärn, "Projective Space in Synthetic Algebraic Geometry"

Abstract: Working in an abstract, homotopy type theory based axiomatization of the higher Zariski-topos called synthetic algebraic geometry, we show that the Picard group of projective n-space is the integers, the automorphism group of projective n-space is PGL(n+1) and morphisms between projective standard spaces are given by homogenous polynomials in the usual way.

https://arxiv.org/abs/2405.13916

=====

Matías Menni, "Bi-directional models of `Radically Synthetic' Differential Geometry", Theory and Applications of Categories, Vol. 40, 2024, No. 15, pp 413-429.

Abstract: The radically synthetic foundation for smooth geometry formulated in [Law11] postulates a space T with the property that it has a unique point and, out of the monoid T^T of endomorphisms, it extracts a submonoid R which, in many cases, is the (commutative) multiplication of a rig structure. The rig R is said to be bi-directional if its subobject of invertible elements has two connected components. In this case, R may be equipped with a pre-order compatible with the rig structure. We adjust the construction of `well-adapted' models of Synthetic Differential Geometry in order to build the first pre-cohesive toposes with a bi-directional R. We also show that, in one of these pre-cohesive variants, the pre-order on R, derived radically synthetically from bi-directionality, coincides with that defined in the original model.
http://www.tac.mta.ca/tac/volumes/40/15/40-15abs.html

highergeometer, 16 days ago to random

If you had to prove there were no square circles, how would you go about it? And state your context and assumptions/definitions up front.

#mtbos

highergeometer, 19 days ago to random

Paiva Miranda De Siqueira, J. V. "Tripos models of Internal Set Theory" (2022)
https://doi.org/10.17863/CAM.78799

This thesis provides a framework to make sense of models of E. Nelson’s Internal Set Theory (and hence of nonstandard analysis) in elementary toposes by exploiting the technology of tripos theory and Lawvere’s hyperdoctrines. A new doctrinal account of nonstandard phenomena is described, which avoids a few key restrictions in Nelson’s approach: chiefly, the dependence on Set Theory (which is done by replacing a model of set theory with a topos as the starting point) and reliance on an internally defined notion of standard element. From the new perspective, validity of the schemes of Idealisation, Standardisation, and Transfer correspond to the existence of certain relationships between hyperdoctrines, leading to the new notion of a tripos model of IST. [... see more in the pdf at the link]

^^^ Thesis supervised by Peter Johnstone and Martin Hyland.

highergeometer, 20 days ago to random

What do you think

"endorsement and initial testing of an AI incubator to enable our transition"

means, in the context of the merger of a pair of universities, and in particular a document talking about curriculum development?

johncarlosbaez, 22 days ago to random

Is there a chance that the physicist Oliver Heaviside was really Wolverine?

image/jpeg

johncarlosbaez, 24 days ago to random

Chemistry is like physics where the particles have personalities - and chemists love talking about the really nasty ones. It makes for fun reading, like Derek Lowe's column "Things I Won't Work With". For example, bromine compounds:

"Most any working chemist will immediately recognize bromine because we don't commonly encounter too many opaque red liquids with a fog of corrosive orange fumes above them in the container. Which is good."

And that's just plain bromine. Then we get compounds like bromine fluorine dioxide.

"You have now prepared the colorless solid bromine fluorine dioxide. What to do with it? Well, what you don't do is let it warm up too far past +10C, because it's almost certainly going to explode. Keep that phrase in mind, it's going to come in handy in this sort of work. Prof. Seppelt, as the first person with a reliable supply of the pure stuff, set forth to react it with a whole list of things and has produced a whole string of weird compounds with brow-furrowing crystal structures. I don't even know what to call these beasts."

https://www.science.org/content/blog-post/higher-states-bromine

johncarlosbaez, 26 days ago (edited 25 days ago) to random

There's a dot product and cross product of vectors in 3 dimensions. But there's also a dot product and cross product in 7 dimensions obeying a lot of the same identities! There's nothing really like this in other dimensions.

We can get the dot and cross product in 3 dimensions by taking the imaginary quaternions and defining

v⋅w= -½(vw + wv), v×w = ½(vw - wv)

We can get the dot and cross product in 7 dimensions using the same formulas, but starting with the imaginary octonions.

The following stuff is pretty well-known: the group of linear transformations of ℝ³ preserving the dot and cross product is called the 3d rotation group, SO(3). We say SO(3) has an 'irreducible representation' on ℝ³ because there's no linear subspace of ℝ³ that's mapped to itself by every transformation in SO(3).

Much to my surprise, it seems that SO(3) also has an irreducible representation on ℝ⁷ where every transformation preserves the dot product and cross product in 7 dimensions!

It's not news that SO(3) has an irreducible representation on ℝ⁷. In physics we call ℝ³ the spin-1 representation of SO(3), or at least a real form thereof, while ℝ⁷ is called the spin-3 representation. It's also not news that the spin-3 representation of SO(3) on ℝ⁷ preserves the dot product. But I didn't know it also preserves the cross product on ℝ⁷, which is a much more exotic thing!

In fact I still don't know it for sure. But @pschwahn asked me a question that led me to guess it's true:

https://mathstodon.xyz/@pschwahn/112435119959135052

and I think I almost see a proof, which I outlined after a long conversation on other things.

The octonions keep surprising me.

https://en.wikipedia.org/wiki/Seven-dimensional_cross_product

highergeometer, 27 days ago to random

Spotted in the wild:

"We understand that a career change is a big decision. This role may even have you considering a move to Adelaide to embark on your next career chapter?"

What's the question?

highergeometer, 1 month ago to random

Oh, cool. Determinants in Japan in the 17thC

"Seki’s approach to “concealed” problems includes a combination of operations he refers to as “interchange” and “cross-multiplication” (kōshiki and shajō, respectively), which effectively involve extracting a determinant. Seki reportedly shared this method in 1683."

https://www.nippon.com/en/japan-topics/c12801/

johncarlosbaez.wordpress.com, 1 month ago to random

https://commons.wikimedia.org/wiki/File:H3_633_FC_boundary.png

This picture by Roice Nelson shows a remarkable structure: the hexagonal tiling honeycomb.

What is it? Roughly speaking, a honeycomb is a way of filling 3d space with polyhedra. The most symmetrical honeycombs are the ‘regular’ ones. For any honeycomb, we define a flag to be a chosen vertex lying on a chosen edge lying on a chosen face lying on a chosen polyhedron. A honeycomb is regular if its geometrical symmetries act transitively on flags.

The most familiar regular honeycomb is the usual way of filling Euclidean space with cubes. This cubic honeycomb is denoted by the symbol {4,3,4}, because a square has 4 edges, 3 squares meet at each corner of a cube, and 4 cubes meet along each edge of this honeycomb. We can also define regular honeycombs in hyperbolic space. For example, the order-5 cubic honeycomb is a hyperbolic honeycomb denoted {4,3,5}, since 5 cubes meet along each edge:

Coxeter showed there are 15 regular hyperbolic honeycombs. The hexagonal tiling honeycomb is one of these. But it does not contain polyhedra of the usual sort! Instead, it contains flat Euclidean planes embedded in hyperbolic space, each plane containing the vertices of infinitely many regular hexagons. You can think of such a sheet of hexagons as a generalized polyhedron with infinitely many faces. You can see a bunch of such sheets in the picture:

https://commons.wikimedia.org/wiki/File:H3_633_FC_boundary.png

The symbol for the hexagonal tiling honeycomb is {6,3,3}, because a hexagon has 6 edges, 3 hexagons meet at each corner in a plane tiled by regular hexagons, and 3 such planes meet along each edge of this honeycomb. You can see that too if you look carefully.

A flat Euclidean plane in hyperbolic space is called a horosphere. Here’s a picture of a horosphere tiled with regular hexagons, yet again drawn by Roice:

Unlike the previous pictures, which are views from inside hyperbolic space, this uses the Poincaré ball model of hyperbolic space. As you can see here, a horosphere is a limiting case of a sphere in hyperbolic space, where one point of the sphere has become a ‘point at infinity’.

Be careful. A horosphere is intrinsically flat, so if you draw regular hexagons on it their internal angles are

2pi/3 = 120^circ

as usual in Euclidean geometry. But a horosphere is not ‘totally geodesic’: straight lines in the horosphere are not geodesics in hyperbolic space! Thus, a hexagon in hyperbolic space with the same vertices as one of the hexagons in the horosphere actually bulges out from the horosphere a bit — and its internal angles are less than 2pi/3: they are

arccosleft(-frac{1}{3}right) approx 109.47^circ

This angle may be familar if you’ve studied tetrahedra. That’s because each vertex lies at the center of a regular tetrahedron, with its four nearest neighbors forming the tetrahedron’s corners.

It’s really these hexagons in hyperbolic space that are faces of the hexagonal tiling honeycomb, not those tiling the horospheres, though perhaps you can barely see the difference. This can be quite confusing until you think about a simpler example, like the difference between a cube in Euclidean 3-space and a cube drawn on a sphere in Euclidean space.

Connection to special relativity

There’s an interesting connection between hyperbolic space, special relativity, and 2×2 matrices. You see, in special relativity, Minkowski spacetime is mathbb{R}^4 equipped with the nondegenerate bilinear form

(t,x,y,z) cdot (t',x',y',z') = t t' - x x' - y y' - z z

usually called the Minkowski metric. Hyperbolic space sits inside Minowski spacetime as the hyperboloid of points mathbf{x} = (t,x,y,z) with mathbf{x} cdot mathbf{x} = 1 and t > 0. But we can also think of Minkowski spacetime as the space mathfrak{h}_2(mathbb{C}) of 2×2 hermitian matrices, using the fact that every such matrix is of the form

A = left( begin{array}{cc} t + z & x - i y \ x + i y & t - z end{array} right)

and

det(A) = t^2 - x^2 - y^2 - z^2

In these terms, the future cone in Minkowski spacetime is the cone of positive definite hermitian matrices:

left{A in mathfrak{h}_2(mathbb{C}) , vert , det A > 0, , mathrm{tr}(A) > 0 right}

Sitting inside this we have the hyperboloid

mathcal{H} = left{A in mathfrak{h}_2(mathbb{C}) , vert , det A = 1, , mathrm{tr}(A) > 0 right}

which is none other than hyperbolic space!

Connection to the Eisenstein integers

Since the hexagonal tiling honeycomb lives inside hyperbolic space, which in turn lives inside Minkowski spacetime, we should be able to describe the hexagonal tiling honeycomb as sitting inside Minkowski spacetime. But how?

Back in 2022, James Dolan and I conjectured such a description, which takes advantage of the picture of Minkowski spacetime in terms of 2×2 matrices. And this April, working on Mathstodon, Greg Egan and I proved this conjecture!

I’ll just describe the basic idea here, and refer you elsewhere for details.

The Eisenstein integers mathbb{E} are the complex numbers of the form

a + b omega

where a and b are integers and omega = exp(2 pi i/3) is a cube root of 1. The Eisenstein integers are closed under addition, subtraction and multiplication, and they form a lattice in the complex numbers:

https://math.ucr.edu/home/baez/mathematical/eisenstein_integers.png

Similarly, the set mathfrak{h}_2(mathbb{E}) of 2×2 hermitian matrices with Eisenstein integer entries gives a lattice in Minkowski spacetime, since we can describe Minkowski spacetime as mathfrak{h}_2(mathbb{C}).

Here’s the conjecture:

Conjecture. The points in the lattice mathfrak{h}_2(mathbb{E}) that lie on the hyperboloid mathcal{H} are the centers of hexagons in a hexagonal tiling honeycomb.

Using known results, it’s relatively easy to show that there’s a hexagonal tiling honeycomb whose hexagon centers are all points in mathfrak{h}_2(mathbb{E}) cap mathcal{H}. The hard part is showing that every point in mathfrak{h}_2(mathbb{E}) cap mathcal{H} is a hexagon center. Points in mathfrak{h}_2(mathbb{E}) cap mathcal{H} are the same as 4-tuples of integers obeying an inequality (the mathrm{tr}(A) > 0 condition) and a quadratic equation (the det(A) = 1 condition). So, we’re trying to show that all 4-tuples obeying those constraints follow a very regular pattern.

Here are two proofs of the conjecture:

• John Baez, Line bundles on complex tori (part 5), The n-Category Café, April 30, 2024.

Greg Egan and I came up with the first proof. The basic idea was to assume there’s a point in mathfrak{h}_2(mathbb{E}) cap mathcal{H} that’s not a hexagon center, choose one as close as possible to the identity matrix, and then construct an even closer one, getting a contradiction. Shortly thereafter, someone on Mastodon by the name of Mist came up with a second proof, similar in strategy but different in detail. This increased my confidence in the result.

What’s next?

Something very similar should be true for another regular hyperbolic honeycomb, the square tiling honeycomb:

https://commons.wikimedia.org/wiki/File:H3_443_FC_boundary.png

Here instead of the Eisenstein integers we should use the Gaussian integers, mathbb{G}, consisting of all complex numbers

a + b i

where a and b are integers.

Conjecture. The points in the lattice mathfrak{h}_2(mathbb{G}) that lie on the hyperboloid mathcal{H} are the centers of squares in a square tiling honeycomb.

I’m also very interested in how these results connect to algebraic geometry! I explained this in some detail here:

• Line bundles on complex tori (part 4), The n-Category Café, April 26, 2024.

Briefly, the hexagon centers in the hexagonal tiling honeycomb correspond to principal polarizations of the abelian variety mathbb{C}^2/mathbb{E}^2. These are concepts that algebraic geometers know and love. Similarly, if the conjecture above is true, the square centers in the square tiling honeycomb will correspond to principal polarizations of the abelian variety mathbb{C}^2/mathbb{G}^2. But I’m especially interested in interpreting the other features of these honeycombs — not just the hexagon and square centers — using ideas from algebraic geometry.

https://johncarlosbaez.wordpress.com/2024/05/04/hexagonal-tiling-honeycomb/

highergeometer, 1 month ago to random

I had a look again today at the proof by Isbell that a skeleton of the category of sets with the cartesian monoidal structure can't be strictly monoidal.

The proof shows that if you assume that the associator (\alpha_{X,X,X}) for an infinite set (X), in a skeleton of (\mathbf{Set}), is the identity arrow, then for every (f\colon X\to X), you have (f=\mathrm{id}_X).

More specifically, it uses the following facts:

1. the universal property of projections
2. the two projection maps (X\times X\to X) are epimorphisms (i.e. onto)
3. Naturality of (\alpha) with respect to the specific morphism ((\mathrm{id}_X,f,\mathrm{id}_X)) in (\mathbf{Set}'\times \mathbf{Set}' \times \mathbf{Set}'), where (\mathbf{Set}') is any skeleton.

In fact, one only needs to use a "local skeleton", namely that (X) is the only set in its isomorphism class. So even just skeletalifying+strictifying a single triple product is impossible.

Exercise: reconstruct the contradiction from these three facts together with the assumption that (\alpha_{X,X,X}=\mathrm{id}).