At this point, I’m sure everyone has seen at least one of the YouTube videos of Hitler ranting (actually actor Bruno Ganz from the movie Downfall) with fake subtitles. Here’s one showing Hitler’s reaction to discovering that in topology a set can be both closed and open. I think we all know how he felt. (This is the clip with accurate subtitles — I’d never seen it before.)
Andrew Gelman quotes from the best possible rejection letter from a journal (sent to Charles Babbage):
It is no inconsiderable degree of reluctance that I decline the offer of any Paper from you. I think, however, you will upon reconsideration of the subject be of the opinion that I have no other alternative. The subjects you propose for a series of Mathematical and Metaphysical Essays are so profound, that there is perhaps not a single subscriber to our Journal who could follow them.
I encourage all journals to adopt this as the standard form letter for rejection.
And you just walked in to find me here with that sad look upon my face.
The computer that hosted Ars Math (which was at a dedicated web-site hosting company) died a horrible death a couple of months ago. For a while, it looked like several years of posts had been lost. They could be recovered by cutting-and-pasting from the Internet Archive, but I found the idea so depressing that I didn’t do anything about recovering the site. Fortunately, we were able to extract the posts from the database anyway. Someone offered to help out with the WordPress hosting, so we’re back online!
At the moment the last couple of years of comments are missing, but we’re still investigating what happened to them.
While economic theory sometimes uses advanced mathematics, such as Brouwer’s fixed point theorem, it’s less common for economic theory to lead to new mathematical developments. The Shapley-Folkman-Starr Theorem is an example of the latter. Roughly, the theorem states that the (Minkowski) sum of a large number of arbitrary sets in a finite-dimensional vector space will be close to convex. Starr was an economics undergraduate who was working on a term paper on approximating non-convex optimization problems with convex ones. This led to collaboration with Shapley (a game theorist), and Folkman (a mathematician), and the eponymous theorem.
Here’s an odd web site. TheoryMine is a site that sells original mathematical theorems. You pay them money, and TheoryMine uses an automated theorem prover to find a theorem and its proof. The paper The Theory Behind TheoryMind explains more about the how it works.
New Scientist has an article with some interviews. The article’s author also bought this theorem.
Machine learning is an exciting new area of applied mathematics and computer science. It’s philosophically interesting because it requires to make a precise mathematical formulation of what it means to learn, and to be able to distinguish concepts from each other. Nils Nilsson has an early draft of a textbook on the subject.
Web comic XKCD has gone bad, and is now telling flagrant and terrible lies about the number line:
Number Line.
(Original here.)
Here‘s a nice little post that explains the significance of positivity in algebraic geometry.
I came across this story about a 19th century swindler who targeted geologists. He would pose (convincingly) as an expert, and exploit the collegiality of geologists to “borrow” money, or valuable specimens, equipment, and books. You would never hear a similar story about mathematicians because they have nothing worth stealing.
Now that I have external evidence that someone is still hoping for new posts, I thought I better write one.
Here’s a result that not only would have I not have guessed, but I would have assumed the opposite is obviously true. There are convex polytopes that cannot be presented in
The survey paper Non-rational configurations, polytopes, and surfaces, by Günter M. Ziegler, gives an explicit construction in 13 dimensions. The paper provides a decent overview of many other, related, topics.