Is it conventional for mathematicians to talk about “the dimensions” like this? I think they are talking about a 126 dimensional space here, but I am a lowly computerer, so maybe this went over my head.
We usually don't talk about "the dimensions", we talk about the general case: n-dimensional spaces (theorems covering all dimensions simultaneously) or infinite-dimensional spaces (individual spaces covering all finite-dimensional spaces).
Of course, when you try to generalize your theorems you are also interested in the cases where generalization fails. In this case, there is something that happens in a 2-dimensional space, in a 6-, 14- or 30-dimensional space. Mathematicians would say "it happens in 2, 6, 14 or 30 dimensions". I never noticed that this is jargon specific to mathematicians.
Problems in geometry tend to get (at least) exponentially harder to solve computationally as the dimensions grow, e.g. the number of vertices of the n-dimensional cube is literally the exponential of base 2. Which is why they discovered something about 126-dimensional space now, when the results for lower dimensions have been known for decades.
But that's not how the article says it. It says "in dimensions 2, 6, 14, 30 and 62" instead of "in 2,6,14 or 30 dimensions". The later sounds fine, but "dimensions 8 and 24" to me sounds too much like something is happening in "8th and 24th dimension". It even uses singular "dimension 126" as if you took >=126 dimensional space, ordered the axis and something interesting happened along 126th and only that one.
Yeah, that's not what that means. In math "dimension" is used as a statistic. As in, "this manifold has a dimension of 4". So you can say things like "in dimension 4" to mean "when the dimension is equal to 4". We do also say "in 4 dimensions"; it just varies. The two phrases are equivalent. There is no ordering of dimensions or anything like that.
you didn't quote the "In". With the "In" it's usual math jargon that means
> "in dimension 4" to mean "when the dimension is equal to 4"
But the title has no "In" and it sounds very weird, perhaps even incorrect. Anyway, note that most of the times the title is not written by the author.
Actually not much of a joke: any kind of array, counting, incremental construction etc. involved in a generic multidimensional space is cut to measure to a certain n value, other basic aspects do not depend on n, leaving out only interesting advanced properties.
It's a good question. It's easy to assume they're talking about R^126 (where R is the reals) but digging a bit deeper I don't think it's true.
The Kervaire invariant is a property of an "n-dimensional manifold", so the paper is likely about 126-dimensional manifolds. That in turn has a formal definition, and although it's not my specialization, I think means it can be locally represented as an n-dimensional Euclidean space.
A simple example would be a circle, which I guess would be a 1-dimensional manifold, because every point on a circle has a tangent where the circle can be approximated by a line passing through the same point.
So they're saying that there are these surfaces which can be locally approximated by 126-dimensional Euclidean spaces. This in turn probably requires that the surface itself is embedded in some higher-dimensional space such as R^127.
Manifolds are generally considered objects of themselves, and it may be difficult to embed then in higher dimensional objects. This is especially the case for tricky manifolds like those with a Kervaire invariant of 1.
Not using the language of this article. Referring to e.g. a two-dimensional space as "Dimension 2" is irregular. One might say that the space has dimension 2 (as shorthand for "has a dimension of 2"), but "Dimension 2" is not used as the proper name of such a space.
It's common in math to say things like "in dimension 2" to mean "when the dimension is 2". It doesn't necessarily refer to a specific space (although it could based on context). It's just setting a contextual variable. Many problems occur in varying dimension and oftentimes you want to restrict discussion to a specific dimension.
Right - what I meant specifically is the use of names like "Dimension 2" (with the capital D) as if to refer to a specific location with that name. Among other things, it has too many associations with pulp science fiction. :)
