User edgar a. bering iv - MathOverflow

How to write popular mathematics well?

2011-12-21T19:04:01Z

Recently, some classmates and I were lamenting the fact that our classmates in other disciplines had almost no conception of what we did, despite the large mathematics population at Waterloo. Instead of giving up in the face of a Very Hard Problem, one of us brought up a column popularizing physics that had a brief run in the school paper, and suggested that we author something similar for mathematics. The column will have some particular constraints that seem challenging to satisfy (self-contained week to week, 500-700 words, try to cover at least some of the current research at UW) but this question is a more general one.

In looking for resources and guidance to help with the writing we have come across several good discussions of topic. We have also found examples of good popular writing and a general discussion of presenting mathematics to a non-mathematical audience.

What we have not found, on MathOverflow or elsewhere, is a popular analogue of the well-answered question "How to write mathematics well?". A lot of the tactical advice of Knuth, Halmos, and others goes out the window when you answer their first question, "Who is your audience?" with "a general university educated public".

What is your advice for writing good mathematics for a popular audience? What holds for all styles of writing and what is article or book specific?

Answer by Edgar A. Bering IV for Excellent uses of induction and recursion

2012-03-30T17:44:34Z

A problem I enjoyed in my undergraduate algorithms course is as follows:

Suppose you have a computing machine with the following architecture. There are $k$ stacks (for some $k$), input can be pushed onto the first stack, output is popped off of the last, and intermediate operations pop from one stack and push to the next in a line. The top of the stack may also be inspected and compared. Given a permutation of ${1,\ldots,n}$ in order as input, how many stacks $k$ do you require to sort the permutation? Describe an algorithm that achieves this bound.

One can prove the bound ($\log_2 n$) by induction, and then just state that this gives a natural recursive algorithm. The same technique was useful for a couple of other problems in a similar vein.

I think this certainly fits the bill of an elegant way to fulfill the task (prove a bound and give an achieving algorithm) in a nice class of cases.

The problem is originally from Knuth Vol. 1, and stack sorting is further elaborated on in this survey.

Computability and complexity of computing $|Hom(G,H)|$ for finitely presented groups G, H.

2011-05-07T16:32:48Z

In the general case, I want to say that determining $|Hom(G,H)|$ is incomputable, arguing that you could use the number to test for simplicity of a presentation, but I am new to this area and I keep finding flaws with my argument.

While I believe the general case is incomputable, there are computable special cases. One in particular that interests me is: compute $|Hom(\pi(S),G)|$ for the fundamental group of a surface $S$, given by a triangulation, and $G$ finite. This arises in Mednykh’s Formula for a 2D TLFT invariant ( $|G|^{\chi(S)-1}|Hom(\pi(S),G)|$), which one can approximate (details in a paper to appear by Gorjan Alagic and myself) efficiently on a quantum computer. However, I have been unable to find any information on the classical complexity of finding $|Hom(\pi(S),G)|$ (with $\pi(S),G$ given in any way) to contrast with the quantum case, or even a discussion of when $|Hom(G,H)|$ is computable and what the complexity of computing it should be.

So, that leaves me with the possibly too broad:

When is $|Hom(G,H)|$ computable for finitely presented $G,H$ and in these special cases what is the classical complexity of computing it?

Answer by Edgar A. Bering IV for What are some examples of journals that will accept undergraduate student research?

2011-05-03T22:55:55Z

I haven't seen mentioned the AMS undergraduate mathematics page, in particular the "Clubs, Conferences, Events, Online Journals" section. The section mentions the following journals targeting undergraduates:

Involve, mentioned in another answer.
The Rose-Hulman Institute of Technology Undergraduate Mathematics Journal, which requires an established mathematician to sponsor your submission and recommend referees.
The Harvard College Mathematics Review, focuses on expository articles by undergraduates. It also sadly seems to be semi-defunct, the website promises an issue in March but one has not yet appeared.

There is also a new undergraduate journal The Waterloo Mathematics Review from the University of Waterloo (full disclosure: I am one of the editors) in a similar style to the HCMR, though it also accepts original research. We are currently accepting submissions for our second issue, while this answer may find you too late I hope you consider submitting.

Comment by Edgar A. Bering IV

2012-02-10T15:35:05Z

Thank you for including publishers in your reply, since that seems to be at the heart of this question.

Comment by Edgar A. Bering IV

2011-05-08T02:23:08Z

Thank you for the pointers. In the case where H is finite this is exponential in the number of generators of $G$, so in the particular case I mentioned above, the genus of the surface $S$ (if one uses the standard presentation for $pi(S)$). Is this the best that can be done classically? In the case where H is solvable the Matei algorithm Eric mentions gives another approach using group cohomology but the complexity isn't clear.