Blog: Articles from May 2010

Shorter entry this week, as I didn’t do much new and exciting in week 2 of my research project. I‘m still having fun, but because it’s so early in the summer, that fun mostly takes the form of reading.

As tweeted earlier, the secret to reading (and understanding) math papers is simple. First, always read it twice. Then read it again. But to make sure you really understand, you need to take notes. Write down what’s implicit in the paper, the steps the author leaves out because “it is obvious” or “it is clear to the reader” or, even worse, “this has been left has an exercise for the reader.” Once you‘ve done that, the final step is to read the paper again.

I spent all week reading two papers, one of which expands on the findings of the other. The first investigates the spreading and covering numbers in relation to the ideal generation conjecture. Much of the paper goes over my head. Nevertheless, there were some very useful figures, and the use of graph theory in one paper and set theory in another helped improve my comprehension of what these numbers are. The second paper, in particular, was devoted to finding explicit values and bounds for the covering number using a combinatorial/set theory approach.

One of my goals is to improve, if I can, upon the bounds found in these papers. The actual values computed by my supervising prof suggest that there’s room for improvement. I’m a little daunted by this prospect. I feel like I understand the proofs present in these two papers regarding the bounds for the covering number … but I‘m not so sure I understand the procedures well enough to build upon them. Granted, I’ve only been doing this for two weeks. As the summer progresses, I’ll learn more and become more confident. For now, however, I’m just a wee bit intimidated by what I will try to accomplish.

Don’t mistake trepidation for discontent. The best is yet to come! Soon I’ll be playing with CoCoA and Macaulay2. This week, I‘m learning about resolution, which leads to a generalizatio of dimension from ordinary vector spaces to modules. Oh, and I’m having a lot of fun learning how to typeset my proofs in LaTeX. Math is totally the language of the universe, and LaTeX is its markup.

I am now into the second week of my NSERC summer research project. So far, I’m having a lot of fun. The subject of my research is interesting and exactly the type of mathematics that I want to study. The “daily grind,” such as it is, does not grind at all—it helps that there are three other undergraduate students doing research this summer, and we all share the sessional lecturer office. We can distract each other, when needed, and pick each other’s brains for help with particularly puzzling proofs.

So what exactly am I doing? Well, it’s esoteric even for those who enjoyed math up until the first years of university. I‘m going to drop some math jargon in the next few paragraphs, so don’t worry if your eyes start to glaze over. Photos and hilarious video will follow!

Since my prof was leaving town at the end of the week, we met several times so he could give me some lectures and we could discuss my project. The work I’m doing relates to ring theory, which is a course I took nearly two years ago, so I have a lot of review to do. Most of the week, like the next few weeks will, involved preliminaries. I found all of the references my prof recommended to me. I began reading the three textbooks among those references, learning about monomial ideals and simplicial complexes.

These, however, are but means to an end. After I have mastered the secrets of these wonderful algebraic concepts, I can use them toward the eventual goal of finding better algorithms for calculating the spreading and covering numbers. These relate to the maximum and minimum dimension, respectively, of a monomial subspace of a vector space over all polynomials of a given degree such that the subspace fulfils two respective properties.

On Thursday, my prof went over what’s changed since he and his colleagues wrote the paper from which my research project comes. In particular, they’ve learned about a connection between edge ideals and the Stanley-Reisner ideal. In the paper, they showed that calculating the dimension of the Stanley-Reisner ring is sufficient to find the spreading number. (A similiar result makes calculating the covering number possible.)

This connection is really cool for two reasons. Firstly, it makes the connection to graph theory stronger, which gives us another avenue for exploring the problem. Secondly, it might provide an alternative way ofcalculating spreading numbers (graph theory is also useful in this respect). The algorithm in the paper finds the Stanley-Reisner ring and then uses a computer algebra system to find the dimension of the ring. They did this on a Pentium II, so they could only find a few of the numbers before the calculations became impractical given the available computer memory. Computing power has improved considerably since then, so my first step will be to see how my little laptop compares against their Pentium II using the algorithm in the paper. Later in the summer, I’ll be creating alternative (hopefully more efficient) algorithms in Macaulay2 and running them on SHARCNET.

Of the three other students sharing the office with me this summer, Aaron is in the same year as me, and Jessica and Rachael are a year behind us. Aaron and Rachel are working on the same project, which involves fractals and Cantor sets. Jessica is also working on something related to commutative algebra (affine varieties and Gröbner bases). So not only do I get to learn about simplicial complexes and monomial ideals, but I’ll be learning about affine spaces and some more real analysis as well.

And for those of you who wonder exactly what math research looks like, I can attest that it’s pretty much like this clip from The Big Bang Theory. Aaron and I spent a good deal of Friday afternoon staring at my faulty proof regarding prime ideals on the chalk board. I did manage to figure it out eventually, but imagine if we had had a montage!

Those of you who read science fiction and fantasy and spend a good deal of time online are probably aware that we’re in the middle of the 2010 Hugo Awards. You can see this year’s nominees here.

While I fall into both of the above categories, I only paid the Hugos passing notice. Certainly, if a book has won the Hugo Award, or even been nominated, then I might give it more consideration before I begin reading it. But not every winner is a winner, if you know what I mean.

This year’s different, though. This year, I’m going to pay more attention, because I’m voting in the Hugo Awards.

Earlier this week, John Scalzi posted on his blog about the 2010 Hugo Voters Packet being available. This is an electronic copy of many of the works nominated for Hugo awards, which is distributed to people who have registered for AussieCon4 (and are thus eligible to vote in the awards).

A full ticket to AussieCon4 is $310 Australian dollars—and I have no intention of attending a convention. But all you need for voting rights is a supporting membership, which is only $70 Australian. I didn’t even need to use my mathematically-inclined brain to figure this one out: for $70, I got DRM-free copies of Hugo-nominated works. This includes all of the nominations for Best Novel. Best Novella, Best Novelette, and Best Short Story. Hard copies of the novels alone would probably set me back more than that (although I already own three of them), and tracking down the short stories, if I were so inclined, would cost even more. Plus, I have a copy of Neil Gaiman’s Batman: Whatever Happened to the Caped Crusader?, which i probably wouldn’t otherwise have read.

Oh, and now I can vote in the Hugos themselves. Which is rather the whole point of this exercise, but I‘m still salivating over all of the new reading material I’ve acquired. My Calibre library has suddenly increased in size, as these works join some public domain books from Project Gutenberg and freebies from Suvudu.

When it comes to this year’s nominees for Best Novel, I’m ahead of the game. As I mentioned, I already own Boneshaker, The City & The City, and Julian Comstock in hard copy. Julian Comstock was one of my top ten books of 2009. I’ve also read and gushed about Wake, so that leaves only two more novels, both of which look superb. I‘m just as interested in seeing which book I choose to support as I am in seeing which one wins.

So anyway, if you want to vote in this year’s Hugo Awards (and nominate works for next year’s Awards) and have $70, head over to the Aussiecon4 website and register. Even if you don’t care to vote, I urge you to check out any of this year’s nominees.

Last summer, the government of Canada held an open consultation on the issue of copyright reform. The result: over 8,300 submissions, over 6,000 of which expressed opposition to another copyright reform bill similar to Bill C-61. You can read my submission here.

It turns out that I and anyone else who submitted to the consultation, wrote a letter to his or her MP, showed up for a meeting or rally, or participated in the Facebook groups or online discourse, have done this all for nothing. We’ve been wasting our time. Because we‘re about to do this all over again.

What’s sad is that it didn’t have to be this way. Tony Clement is the Industry Minister now, and his attitude toward copyright reform is more sensible than Jim Prentice’s. Apparently he was open to a different approach than the one Bill C-61 took—and considering how unacceptable Bill C-61 was, I’ll take that. Alas, it looks like Mr. Clement and his fellow cabinet minister, James Moore have differing opinions. So Grandfather Harper intervened.

The result will apparently be a “Canadian DMCA” that is, as Cory Doctorow puts it, a “goddamned disaster.” While I don’t know if the new bill will truly live up to such rhetoric, I’m certain that a repeat of Bill C-61 is something Canadians neither want nor need. Not only does it mean that the copyright consultation was a huge waste of government time and taxpayer money, but it means the government refuses to listen to the people who elected it. Again.

I don’t know about you, but I’m starting to see a trend: proguing Parliament, denying access to the Afghan detainee documents, ignoring the results of a national copyright consultation … time and again, the Harper government had demonstrated that it is unwilling to listen to the public and prefers secrecy over transparency.

Michael Geist calls on us to “write a paper letter to your Member of Parliament” to express our dissatisfaction with this turn of events. I might have done that last year, but now my idealism is beginning to crack and peel. My MP is a member of the NDP! What change will he be able to effect? Even if the NDP hadn’t expressed support for fair copyright reform, they’d probably oppose the Conservatives out of political need anyway. But as long as the Harper government remains in power, the opposition parties will always be on the defensive.

Despite their sabre-rattling, do any of our opposition leaders really want an election? I don’t blame them for wanting to wait and see—none of our leaders impress me right now; my strategy at this point is pretty much, “don’t vote Conservative.” It’s not that I’m gung ho to see another Liberal regime … we just don’t really have many other options. Michael Ignatieff was in Thunder Bay yesterday to announce universal broadband access to all Canadians—especially rural Canadians—as part of the Liberal platform. Don’t get me wrong; I think it’s a wonderful idea and fully support it—but it’s just an idea right now. They’ve been talking about improving broadband access for years. I’ll believe it when I see it happen.

What I do believe is that a “Canadian DMCA” of any kind is a bad idea. However, it looks like as long as Harper is in the driver’s seat—with Moore riding shotgun—that will be the only option on the table. And to that I say: I‘m done.

You win, Mr. Harper. Please, do continue to prorogue Parliament and bully your MPs. Please, do continue to flout the democratic principles upon which our country was founded and reshape Canada into your perfect little principality. I will meekly go back to my books, stick my head in the sand, and exude the level of apathy stereotypical of my demographic.

Wake me up when there’s an election.