While I understand manifolds, this article is the first time I have seen the term "contact manifold." Does anyone know of an original source where "contact manifold" is first formally defined?
Not an original source, but the Wikipedia page [0] gives a good introduction. A contact manifold is then just a manifold equipped with a contact structure.
A more in-depth discussion can be found in many differential topology texts, e.g. page 581 of [1].
Seems solid enough. There's lots of other fun links there. I'm fascinated by this field but I don't have a degree of any kind so ... lots more learning to go!!!
but when I understand manifolds both contermporary flat-earthers and scientific academics(?) get mad at me?
but I wouldn't claim to understand manifolds, I just understand that they have something to do with flatness perception at small scales (or short distances)
there are two types of mathematics: cutting-edge stuff like this ,and then the stuff that fills textbooks, random arxiv pre-prints, etc that is more mundane
I'm not sure what you mean by that. The "stuff that fills textbook" is at least a century or two old. "Random arXiv preprints" are (usually) literally cutting-edge research.
I wouldn't necessarily discount the value of the older maths; I'm taking analysis and abstract algebra as my first classes in a maths masters program; most everything is from the 1800s or early 1900s but is still mind blowingly cool and not necessarily well known on the street. Vitali sets, the Burnside lemma, group actions, all quite fun. And this is before even treating a function space as a vector space or Galois theory.
That is literally my point restated. mathematicians who write textbooks are covering stuff which is old, not cutting edge. most arxiv posts are not cutting edge, only a tiny percentage are, like the stuff mentioned here.
That's absurd. I'm a math researcher. I follow a couple of arxiv categories. Almost everything on there is new theorems. That's the definition of cutting edge.
> there are two types of mathematics: cutting-edge stuff like this ,and then the stuff that fills textbooks, random arxiv pre-prints, etc that is more mundane
I somewhat disagree.
First: obviously, if you want to read really cutting edge stuff, in nearly all cases you have to read papers or preprints.
On the other hand, there do exist various kinds of math books: in rough decreasing order of topicality and increasing order of understandability for "ordinary" mathematicians (i.e. not specialists in the respective field):
- collections of recent research papers
- research monographs
- survey monographs/survey collections about some active research topic
I think your picture is slightly inaccurate. It is more like, there are the random arxiv pre-prints, then 1 in 1000 of those get featured in popular press like this or are otherwise truly remarkable, and 1 in 100 of those make it into the textbooks.
The Gauss and Euler results you find in textbooks are not mundane, they are the "arxiv pre-prints" of 200 years ago that made it through the filter of time because they turned out to be important or deep.
