Feeds:
Posts

## Writing a blog post every day is hard and possibly not a good idea

So: I’m happy that I’ve kept up MaBloWriMo for 13 days so far, but I’m running out of steam. I’ve gone through essentially all of the posts in my backlog that were relatively easy to write, and the things I’d like to write about at this point either

• really should be done with diagrams (and it’s not easy to finish a blog post with diagrams in a day) or
• might take more time than I allot for blogging in a day to work through the relevant concepts.

Sticking to one post a day at this point is likely to drive down quality, so I think I am going to stop doing it. It was a good goal for awhile in that it got me to write some posts that I’d wanted to write for a long time now, but unfortunately it is now doing the opposite of that.

## MaBloWriMo is upon us

Three years ago I thought it would be fun to write a blog post every day of November. I’m not sure why I didn’t do this in November 2010 or 2011 because I’m pretty sure I learned a lot from doing it in 2009, so I’d like to do it again. The posts will probably be shorter this time.

## Morality

Apologies for the lack of updates; I’ve been attempting to apply to graduate school. In the meantime, I want to link to a fantastic paper I just heard about by Eugenia Cheng on moral truth in mathematics. In private (or for me, on MathOverflow), mathematicians often say things like “well, morally, this should be true because…” and Cheng extensively discusses what this could mean and why it’s important.

I’m glad I finally have a word for this. I’ve cared about moral truth more than proof for awhile now, and that’s a major reason I’ve been trying to teach myself physics: even if it isn’t a good source of proofs, it seems like a great source of moral truths.

## Update

Apologies for the lack of posts recently; I’m on Spring Break and trying to relax and get some work done instead of blogging.

In other news, Stack Exchange offered me, and I accepted, an applied-mathematics internship with them this summer. (Procrastinating on math.SE finally paid off!)

I have learned some interesting things about mathematics education from seeing the types of questions which pop up most commonly on math.SE. If I can think of a way to coherently summarize my thoughts I might write a post about it.

## Pi is still wrong

In anti-honor of “Pi Day,” I’d like to direct your attention to Michael Hartl’s The Tau Manifesto. The Manifesto is inspired by Bob Palais’ article $\pi$ is wrong! and presents a list of simple, but compelling, reasons that $2 \pi$, not $\pi$, is the more fundamental constant.

These ideas have been discussed on the blathosphere before, e.g. on Bill Gasarch and Lance Fortnow’s blog Computational Complexity. There Terence Tao makes the following remark:

It may be that $2 \pi i$ is an even more fundamental constant than $2 \pi$ or $\pi$. It is, after all, the generator of $\log(1)$. The fact that so many formulae involving $\pi^n$ depend on the parity of $n$ is another clue in this regard.

The basic argument for this point of view can be summarized as follows: $e^z : \mathbb{C} \to \mathbb{C}$ is a special function because it is the unique eigenvector of eigenvalue $1$ of the derivative operator acting on, say, complex-analytic functions on $\mathbb{C}$, and this function has period $2 \pi i$. So we see that this constant pops directly out of a definition of $\mathbb{C}$ and a definition of the derivative of a complex-analytic function: no arbitrary choices were necessary. (The closest thing to an arbitrary choice here is the decision to identify the tangent space of a point in a vector space with the vector space itself, but this is completely invariant.)

The $2 \pi$ here is precisely the circumference of a unit circle, which is distinguished among all circles because in $\mathbb{C}$ it is the only circle of positive radius closed under multiplication. This is a fundamental number because of the relationship between the unit circle and Pontrjagin duality (which has the Fourier transform and Fourier series as special cases), and is responsible for all appearances of $2 \pi$ in mathematics that I know of.

For example, the reason there is a factor of $\sqrt{2\pi}$ in the definition of the Gaussian distribution (which is where the factor of $\sqrt{2\pi}$ comes from in Stirling’s formula) is that the Gaussian distribution is its own Fourier transform. This factor is commonly cited as an application of $\pi$ that has nothing to do with circles, but of course the Fourier transform has everything to do with circles.

Edit, 3/15/11: Vi Hart also explains the wrongness of $\pi$ in video form. I have to admit I think I read the title of her post and then promptly forgot I had done so when writing this post.

## John Baez interviews Eliezer Yudkowsky

There’s a lot of food for thought in John Baez’s latest post on Azimuth, an interview with AI researcher Eliezer Yudkowsky.

Eliezer Yudkowsky happens to be one of the most interesting people I know of. In addition to his work on friendly AI, he helped found the community blog Less Wrong. The material in his Sequences there describes what Yudkowsky calls “the art of rationality,” but if you’re not up to reading several long sequences of blog posts, you might be interested in his enormously popular Harry Potter fanfic, Harry Potter and the Methods of Rationality, which explores many of the same ideas in a more fun and accessible setting. I think this was an enormously clever ploy on his part.

Since this is a math blog, I’d also like to highlight the following part of the interview above:

In Silicon Valley a failed entrepreneur still gets plenty of respect, which Paul Graham thinks is one of the primary reasons why Silicon Valley produces a lot of entrepreneurs and other places don’t. Robin Hanson is a truly excellent cynical economist and one of his more cynical suggestions is that the function of academia is best regarded as the production of prestige, with the production of knowledge being something of a byproduct. I can’t do justice to his development of that thesis in a few words (keywords: hansom academia prestige) but the key point I want to take away is that if you work on a famous problem that lots of other people are working on, your marginal contribution to human knowledge may be small, but you’ll get to affiliate with all the other prestigious people working on it.

Words to ponder in light of the refusal of famous mathematicians like Grothendieck and Perelman to associate with academia.

Edit, 3/14/11: Part two of the interview is up.

## The 2010 Fields Medalists

The 2010 Fields Medalists have just been announced! The moment I found their names on Twitter, they were already on the Wikipedia article. Go figure.

I am not familiar with any of their names, except that Ngô Bảo Châu’s name was tossed around on the blogosphere awhile back because of his proof of the fundamental lemma.

I’ve been reading a lot of mathematics lately, but I don’t feel capable of explaining most of what I’ve been reading about, so I’m not sure what to blog about these days. Fortunately, SPUR will be starting soon, so I’ll start focusing on relevant material for my project eventually. Until then, here are some more random updates.

• Martin Gardner and Walter Rudin both recently passed away. They will be sorely missed by the mathematical community, although I can’t say I’m particularly qualified to eulogize about either.
• For my number theory seminar with Scott Carnahan I wrote a paper describing an important corollary of the Eichler-Shimura relation in the theory of modular forms. The actual relation is somewhat difficult to state, but the important corollary relates the number of points on certain elliptic curves $E$ over finite fields to the Fourier coefficients of certain modular forms of weight $2$. You can find the paper here. Although the class is over, corrections and comments are of course welcome. (Though I hope Scott doesn’t change my grade if someone spots a mistake he missed!)
• If you’re at all interested in the kind of mathematics where planar diagrams are used instead of traditional algebraic notation for computation, you should read Joachim Kock’s excellent book Frobenius Algebras and 2D Topological Quantum Field Theories. The book is much less intimidating than its title might suggest, and it is full of enlightening pictures and discussions. You might also be interested in a related MO question.

I won’t have time for a substantive update for about a week, so here are some bullet points.

• Peter Cameron has a blog, which I somehow didn’t know before. Terence Tao and Tim Gowers both have links to it, but the Secret Blogging Seminar doesn’t. Anyway, it’s excellent.
• I’ve been looking over the course listings at Cambridge for Part II. Conclusion: by Cambridge standards, I ought to know more physics.
• Andrea Ferretti recently started a new math website, MathOnline. It’s for posting links to free mathematics resources on the internet. I think all of my favorite online resources are already on it, so I haven’t contributed anything yet, but I’m happy that they’re all gathered in one place.
• I’ve also been reading Kassel’s Quantum Groups and Turaev’s Quantum Invariants of Knots and 3-Manifolds, in part to better understand Scott Morrison’s answer to my question about the Catalan numbers, but mostly because what little I know about the subject is fascinating. But I feel guilty about doing this; I can’t help but feel that as an undergraduate I should be paying more attention to my fundamentals and learning about fancy stuff later. What do you think?

## Textbooks

I recently added two new pages to the blog: a bibliography for listing references I cite on multiple occasions, and a suggestions and requests page. The bibliography is likely to soon contain citations for at least some of the following books which have recently come into my possession:

1. Introduction to the Theory of Computation, Sipser
2. Lectures on Quantum Mechanics, Faddeev, Yakubovskii
3. Representation Theory: a First Course, Fulton, Harris
4. Conceptual Mathematics, Lawvere, Schanuel
5. Concrete Mathematics: a Foundation for Computer Science, Graham, Knuth, Patashnik

I haven’t looked at 2 or 4 very closely yet, but so far I find 1, 3, and 5 to be among the best written textbooks I have ever read. Sipser’s book, in particular, strikes me as having found a perfect balance between brevity and clarity. His tone is conversational but finely polished, and I rather like his habit of summarizing the basic strategy of a proof before actually writing it down. Generally I am finding the book an absolute pleasure to read, which I can’t say for most of the math textbooks I’ve seen. You will likely see me blogging a little about languages and automata once I finish up my current series (right now I’m stuck on what should be a trivial proof).

Why don’t more mathematicians write like Sipser?