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March 14, 2015

Split Octonions and the Rolling Ball

Posted by John Baez

You may enjoy these webpages:

because they explain a nice example of the Erlangen Program more tersely — and I hope more simply — than before, with the help of some animations made by Geoffrey Dixon using WebGL. You can actually get a ball to roll in way that illustrates the incidence geometry associated to the exceptional Lie group G 2\mathrm{G}_2!

Abstract. Understanding exceptional Lie groups as the symmetry groups of more familiar objects is a fascinating challenge. The compact form of the smallest exceptional Lie group, G 2\mathrm{G}_2, is the symmetry group of an 8-dimensional nonassociative algebra called the octonions. However, another form of this group arises as symmetries of a simple problem in classical mechanics! The space of configurations of a ball rolling on another ball without slipping or twisting defines a manifold where the tangent space of each point is equipped with a 2-dimensional subspace describing the allowed infinitesimal motions. Under certain special conditions, the split real form of G 2\mathrm{G}_2 acts as symmetries. We can understand this using the quaternions together with an 8-dimensional algebra called the ‘split octonions’. The rolling ball picture makes the geometry associated to G 2\mathrm{G}_2 quite vivid. This is joint work with James Dolan and John Huerta, with animations created by Geoffrey Dixon.

Posted at 1:18 AM UTC | Permalink | Followups (12)

March 11, 2015

A Scale-Dependent Notion of Dimension for Metric Spaces (Part 1)

Posted by Simon Willerton

Consider the following shape that we are zooming into and out of. What dimension would you say it was?

zooming dots

At small scales (or when it is very far away) it appears as a point, so seems zero-dimensional. At bigger scales it appears to be a line, so seems one-dimensional. At even bigger scales it appears to have breadth as well as length, so seems two-dimensional. Then, finally, at very large scales it appears to be made from many widely separated points, so seems zero-dimensional again.

We arrive at an important observation:

The perceived dimension varies with the scale at which the shape is viewed.

Here is a graph of my attempt to capture this perceived notion of dimension mathematically.

A graph

Hopefully you can see that, moving from the left, you start off at zero, then move into a region where the function takes value around one, then briefly moves up to two then drops down to zero again.

The ‘shape’ that we are zooming into above is actually a grid of 3000×163000\times 16 points as that’s all my computer could handle easily in making the above picture. If I’d had a much bigger computer and used say 10 7×10 210^{7}\times 10^{2} points then I would have got something that more obviously had both a one-dimensional and two-dimensional regime.

In this post I want to explain how this idea of dimension comes about. It relies on a notion of ‘size’ of metric spaces, for instance, you could use Tom’s notion of magnitude, but the above picture uses my notion of 22-spread.

I have mentioned this idea before at the Café in the post on my spread paper, but I wanted to expand on it somewhat.

Posted at 3:25 PM UTC | Permalink | Followups (19)

March 9, 2015

HITs and the Erlangen Program

Posted by Mike Shulman

Last week I had a minor epiphany, which I would like to share. If it holds up, it’ll probably end up somewhere in my chapter for Landry’s book.

I was fortunate during my formal education to have the opportunity to take a fair number of physics classes in addition to mathematics. Over that time I was introduced to tensors (more specifically, tensor fields on manifolds) several times in a row: first in special relativity, then in differential geometry, and then again in general relativity and Riemannian geometry.

During all those classes, I remember noticing that mathematicians and physicists tended to speak about tensors in different ways. To a mathematician, a tensor (field) was a global geometric object associated to a manifold; upon choosing a local chart it could be expressed using local coordinates, but fundamentally it was a geometric thing. By contrast, physicists tended to talk about a tensor as a collection of coordinates labeled by indices, with its “tensorial” nature encoded in how those coordinates transformed upon changing coordinates.

(To be sure, most physicists nowadays are aware of the mathematician’s perspective, and many even use it. But I still got the sense that they found it easier, or at least expected that we students would find it easier, to think in coordinates.)

Here is my epiphany: the symbiosis between these two viewpoints on tensors is closely related to the symbiosis between HITs and univalence in homotopy type theory. I will now explain…

Posted at 7:21 AM UTC | Permalink | Followups (14)

March 5, 2015

Mathematics Without Apologies

Posted by David Corfield

Since I had reviewed the manuscript for Princeton University Press eighteen months ago, this week I received my complementary copy of Mathematics Without Apologies by Michael Harris.

Michael, as most people here will know, is a number theorist whose successes include, with Richard Taylor, the proof of the local Langlands conjecture for the general linear group GL n(K)GL_n(K) in characteristic 0. But alongside being a prize-winning mathematician, he also likes to think hard about the nature of mathematics. He spoke at a conference I co-organised, Two Streams in the Philosophy of Mathematics, and I’ve met up with him on a number of other occasions, including the Delphi meeting with John. He’s extremely well placed then to give an account of the life of a current mathematician with, as suggested by the book’s subtitle, Portrait of a problematic vocation, all its peculiarities.

I’ll be re-reading the book once term is over, and say more then, but for now those wanting to find out more can read a Q&A with the author, and drafts of some chapters available here. Michael has also set up an associated blog, named after the book.

Posted at 9:09 AM UTC | Permalink | Followups (3)

February 27, 2015

Concepts of Sameness (Part 4)

Posted by John Baez

This time I’d like to think about three different approaches to ‘defining equality’, or more generally, introducing equality in formal systems of mathematics.

These will be taken from old-fashioned logic — before computer science, category theory or homotopy theory started exerting their influence. Eventually I want to compare these to more modern treatments.

If you know other interesting ‘old-fashioned’ approaches to equality, please tell me!

Posted at 10:21 AM UTC | Permalink | Followups (32)

February 26, 2015

Introduction to Synthetic Mathematics (part 1)

Posted by Mike Shulman

John is writing about “concepts of sameness” for Elaine Landry’s book Category Theory for the Working Philosopher, and has been posting some of his thoughts and drafts. I’m writing for the same book about homotopy type theory / univalent foundations; but since HoTT/UF will also make a guest appearance in John’s and David Corfield’s chapters, and one aspect of it (univalence) is central to Steve Awodey’s chapter, I had to decide what aspect of it to emphasize in my chapter.

My current plan is to focus on HoTT/UF as a synthetic theory of \infty-groupoids. But in order to say what that even means, I felt that I needed to start with a brief introduction about the phrase “synthetic theory”, which may not be familiar. Right now, my current draft of that “introduction” is more than half the allotted length of my chapter; so clearly it’ll need to be trimmed! But I thought I would go ahead and post some parts of it in its current form; so here goes.

Posted at 6:15 AM UTC | Permalink | Followups (50)

February 25, 2015

Concepts of Sameness (Part 3)

Posted by John Baez

Now I’d like to switch to pondering different approaches to equality. (Eventually I’ll have put all these pieces together into a coherent essay, but not yet.)

We tend to think of x=xx = x as a fundamental property of equality, perhaps the most fundamental of all. But what is it actually used for? I don’t really know. I sometimes joke that equations of the form x=xx = x are the only really true ones — since any other equation says that different things are equal — but they’re also completely useless.

But maybe I’m wrong. Maybe equations of the form x=xx = x are useful in some way. I can imagine one coming in handy at the end of a proof by contradiction where you show some assumptions imply xxx \ne x. But I don’t remember ever doing such a proof… and I have trouble imagining that you ever need to use a proof of this style.

If you’ve used the equation x=xx = x in your own work, please let me know.

Posted at 1:20 AM UTC | Permalink | Followups (39)

February 23, 2015

Concepts of Sameness (Part 2)

Posted by John Baez

I’m writing about ‘concepts of sameness’ for Elaine Landry’s book Category Theory for the Working Philosopher. After an initial section on a passage by Heraclitus, I had planned to write a bit about Gongsun Long’s white horse paradox — or more precisely, his dialog When a White Horse is Not a Horse.

However, this is turning out to be harder than I thought, and more of a digression than I want. So I’ll probably drop this plan. But I have a few preliminary notes, and I might as well share them.

Posted at 3:31 AM UTC | Permalink | Followups (49)

Concepts of Sameness (Part 1)

Posted by John Baez

Elaine Landry is a philosopher at U. C. Davis, and she’s editing a book called Categories for the Working Philosopher. Tentatively, at least, it’s supposed to have chapters by these folks

  • Colin McLarty (on set theory)
  • David Corfield (on geometry)
  • Michael Shulman (on univalent foundations)
  • Steve Awodey (on structuralism, invariance, and univalence)
  • Michael Ernst (on foundations)
  • Jean-Pierre Marquis (on first-order logic with dependent sorts)
  • John Bell (on logic and model theory)
  • Kohei Kishida (on modal logic)
  • Robin Cockett and Robert Seely (on proof theory and linear logic)
  • Samson Abramsky (on computer science)
  • Michael Moortgat (on linguistics and computational semantics)
  • Bob Coecke and Aleks Kissinger (on quantum mechanics and ontology)
  • James Weatherall (on spacetime theories)
  • Jim Lambek (on special relativity)
  • John Baez (on concepts of sameness)
  • David Spivak (on mathematical modeling)
  • Hans Halvorson (on the structure of physical theories)
  • Elaine Landry (on structural realism)
  • Andrée Ehresmann (on a topic to be announced)

We’re supposed to have our chapters done by April. To make writing my part more fun, I thought I’d draft some portions here on the nn-Café.

Posted at 12:20 AM UTC | Permalink | Followups (30)

February 18, 2015

Quantum Physics and Logic at Oxford

Posted by John Baez

There’s a workshop on quantum physics and logic at Oxford this summer:

Posted at 9:52 PM UTC | Permalink | Followups (1)

February 12, 2015

Can a Computer Solve Lebesgue’s Universal Covering Problem?

Posted by John Baez

Here’s a problem I hope we can solve here. I think it will be fun. It involves computable analysis.

To state the problem precisely, recall that the diameter of a set of points AA in a metric space is

diam(A)=sup{d(x,y):x,yA} diam(A)=\sup\{d(x,y) : x,y\in A\}

Recall that two subsets of the Euclidean plane 2\mathbb{R}^2 are isometric if we can get one from the other by translation, rotation and/or reflection.

Finally, let’s define a universal covering to be a convex compact subset KK of the Euclidean plane such that any set A 2A \subseteq \mathbb{R}^2 of diameter 11 is isometric to a subset of KK.

In 1914 Lebesgue posed the puzzle of finding the universal covering with the least area. Since then people have been using increasingly clever constructions to find universal coverings with smaller and smaller area.

My question is whether we need an unbounded amount of cleverness, or whether we could write a program to solve this puzzle.

There are actually a number of ways to make this question precise, but let me focus on the simplest. Let 𝒰\mathcal{U} be the set of all universal coverings. Can we write a program that computes this number to as much accuracy as desired:

a=inf{area(K):K𝒰}? a=inf\{ area(K) : K \in \mathcal{U}\} \; ?

More precisely, is this real number aa computable?

Right now all we know is that

0.832a0.844115376859 0.832 \le a \le 0.844115376859\dots

though Philip Gibbs has a heuristic argument for a better lower bound:

0.84408a 0.84408 \le a

Posted at 12:53 AM UTC | Permalink | Followups (45)

February 9, 2015

Higher-Dimensional Rewriting in Warsaw

Posted by John Baez

This summer there will be a conference on higher-dimensional algebra and rewrite rules in Warsaw. They want people to submit papers! I’ll give a talk about presentations of symmetric monoidal categories that are important in electrical engineering and control theory. There should also be interesting talks about combinatorial algebra, homotopical aspects of rewriting theory, and more:

Here’s a description…

Posted at 10:36 PM UTC | Permalink | Post a Comment

February 4, 2015

More on the AMS and NSA

Posted by Tom Leinster

Just a quickie. This month’s Notices of the AMS ran an article by Michael Wertheimer, recently-retired Director of Research at the NSA, largely about the accusation that the NSA deliberately created a backdoor in a standard cryptographic utility so that they could decode the messages of anyone using it.

Wertheimer’s protestations garnered an unusual amount of press and a great deal of scepticism (e.g. Le Monde, Ars Technica, The Register, Peter Woit, me), with the scepticism especially coming from crypto experts (e.g. Matthew Green, Ethan Heilman).

Some of those experts — also including Bruce Schneier — are writing to the Notices pointing out how misleading Wertheimer’s piece was, with ample historical evidence. And crucially: that in everything Wertheimer wrote, he never actually denied that the NSA created a backdoor.

If you support this letter — and if more broadly, you think it’s important that the AMS reconsiders its relationship with the NSA — then you can add your signature.

Posted at 2:37 AM UTC | Permalink | Followups (9)

Lebesgue’s Universal Covering Problem

Posted by John Baez

Lebesgue’s universal covering problem is famously difficult, and a century old. So I’m happy to report some progress:

• John Baez, Karine Bagdasaryan and Philip Gibbs, Lebesgue’s universal covering problem.

But we’d like you to check our work! It will help if you’re good at programming. As far as the math goes, it’s just high-school geometry… carried to a fanatical level of intensity.

Posted at 1:23 AM UTC | Permalink | Followups (12)

January 19, 2015

The Univalent Perspective on Classifying Spaces

Posted by Mike Shulman

I feel like I should apologize for not being more active at the Cafe recently. I’ve been busy, of course, and also most of my recent blog posts have been going to the HoTT blog, since I felt most of them were of interest only to the HoTT crowd (by which I mean, “people interested enough in HoTT to follow the HoTT blog” — which may of course include many Cafe readers as well). But today’s post, while also inspired by HoTT, is less technical and (I hope) of interest even to “classical” higher category theorists.

In general, a classifying space for bundles of XX’s is a space BB such that maps YBY\to B are equivalent to bundles of XX’s over YY. In classical algebraic topology, such spaces are generally constructed as the geometric realization of the nerve of a category of XX’s, and as such they may be hard to visualize geometrically. However, it’s generally useful to think of BB as a space whose points are XX’s, so that the classifying map YBY\to B of a bundle of XX’s assigns to each yYy\in Y the corresponding fiber (which is an XX). For instance, the classifying space BOB O of vector bundles can be thought of as a space whose points are vector spaces, where the classifying map of vector bundle assigns to each point the fiber over that point (which is a vector space).

In classical algebraic topology, this point of view can’t be taken quite literally, although we can make some use of it by identifying a classifying space with its representable functor. For instance, if we want to define a map f:BOBOf:B O\to B O, we’d like to say “a point vBOv\in B O is a vector space, so let’s do blah to it and get another vector space f(v)BOf(v)\in B O. We can’t do that, but we can do the next best thing: if blah is something that can be done fiberwise to a vector bundle in a natural way, then since Hom(Y,BO)Hom(Y,B O) is naturally equivalent to the collection of vector bundles over YY, our blah defines a natural transformation Hom(,BO)Hom(,BO)Hom(-,B O) \to Hom(-,B O), and hence a map f:BOBOf:B O \to B O by the Yoneda lemma.

However, in higher category theory and homotopy type theory, we can really take this perspective literally. That is, if by “space” we choose to mean “\infty-groupoid” rather than “topological space up to homotopy”, then we can really define the classifying space to be the \infty-groupoid of XX’s, whose points (objects) are XX’s, whose morphisms are equivalences between XX’s, and so on. Now, in defining a map such as our ff, we can actually just give a map from XX’s to XX’s, as long as we check that it’s functorial on equivalences — and if we’re working in HoTT, we don’t even have to do the second part, since everything we can write down in HoTT is automatically functorial/natural.

This gives a different perspective on some classifying-space constructions that can be more illuminating than a classical one. Below the fold I’ll discuss some examples that have come to my attention recently.

Posted at 6:25 PM UTC | Permalink | Followups (15)