Free time to work

I have the quarter off from teaching, which means it’s the perfect time to get some serious work done on my thesis. The fact that the work that it most important in my life is also the least urgent—I could do nothing for six months and no one would know or care—means I have to be especially disciplined with myself. To that end, I’m putting aside big tracts of time that are to be devoted to my work.

I feel like I’ve finally started to get the good work habits and general resolve in place. Becoming disciplined in this way was one of my original goals when I came to graduate school.

Unfortunately, the work that I’m looking at right now is in danger of devolving into long strings of algebraic manipulation with no clear end in sight. There’s a difference between devloting time to working and working well. With math as with so many things, the best flow opens up when you’re on a certain kind of a roll. Sadly, there seems to be so much to interrupt said roll in mathematics: so many slippery details that get away from you.

I had a professor once say that you only do an hour of work in graduate school: it just needs two years of preparation to make it happen.

Beauty Through Mathematics

Link: Beauty Through Mathematics

Here’s another group out to show the world a vision of beauty through mathematics: Imaginary. Computers have made wonderful things possible in the realm of visualization. It is so labor intensive to get pictures this good without them.

Also, on the topic of mathematics having it’s moment in the world—Germany just named 2008 the Year of Mathematics. Every dog has its day.

My favorite symbol

Earlier this year a middle schooler emailed me to ask me what my favorite mathematical symbol was. I didn’t have a good answer, because I’d never thought about it at length, and I ended up giving him the old standby answer: 0. The apparent contradictory nature of having something stand for nothing kept humanity from developing it for some time, and its existence allows numbers to be written in the modern base 10 (or base anything) form. Books have been written about the innovation, like this one, or this one.

Honestly, though, I don’t have much of a personal connection to zero. The symbol I find really compelling, though, is the arrow. Let me explain why.

First of all, it’s everywhere. I can’t think of a mathematical subfield that doesn’t involve drawing arrows (each with their own precise mathematical meaning). We call them morphisms, mappings, functions, edges of directed grahps, etc., but it seems like they’re always present. The idea of moving your problem from here to there is central in mathematics.

Second, I love the implication of movement. There’s a line about chess, that “the threat is more powerful than the move.” Understanding how to look at a chess board as a place where forces are exerting themselves represents a leap forward in one’s ability with the game. Similarly, feeling a mathematical situation as wanting to be represented elsewehre is a key part of a developed (or developing) mathematical intuition.

Third, it’s powerful. It’s shocking how much information is carried in arrowed diagrams. They allow you to ignore clutter and focus just on what is essential. Category theory is a dramatic example of this—everything is reduced to objects and the arrows between them.

So, to that middle schooler I led astray with an out-of-date, impersonal answer, I hope you find this post.

the moment

There’s a line in Brecht’s Galileo where the eponymous character says, and I paraphrase, that in his lifetime he saw astronomy come out into the public sphere in a way it never had before. I believe this is such a moment for mathematics, and I also believe that it’s on the way out.

Invisibility

Link: Invisibility

I sometimes have to defend math in the world. Normally, I follow Hardy and say that the reason to do math is not because it’s useful but because it’s beautiful. Every once in a while, though, it’s worth bringing up an application because it’s just so cool. Gunther Uhlmann in my department is on a team that’s closing in on making the cloak of invisibility a reality.

My understanding is that Gunther works on inverse problems, such as the one that oil companies are very interested in: if you have, say, sonar readings of an undersea or underground regions, can you figure out what’s down there? Gunther was working on general versions of the problem and noticed that under certain conditions the readings would totally miss pockets underground. And the same principle, perhaps, can be extended to light waves bouncing off objects. Pretty amazing stuff.

Still Here!

Been busy lately with teaching and other stuff, but I’m still here and blogging. And, a fellow grad just gave me a copy of his manuscript on math in his life (I plan to write a very similar book sometime), so I’ll be reading that and writing about it soon. But, I have to give a final to my students tomorrow, and it’s very important to calibrate the difficulty of these things correctly. Stay tuned!

Update: fluther solved my problem

I wasn’t expecting it, but someone on fluther managed to answer my question (see last post). Now I’m in an interesting position: I have a hunch that whenever this certain algebraic equation has nontrivial solutions, that corresponds to the geometric objects I’m studying being the same as each other. But there doesn’t seem to be any clear reason this would be true. As always, I need to know more. There’s a paper I’m going to dive in to, which might have the answer, or point me on the right path.

Algebraic Geometry Conference Week 2

First of all, someone commented that it’s not really a conference, since it lasts for six months. More like a program. I’ve been in a tiny bit of a conundrum when talk times roll around: do I attend the talks, or do I keep at my own work, where I’ve built up some momentum? Or to I go to the talk and surreptitiously keep doing my own work? That last option was the one I often took, but the last two days I opted to avoid MSRI altogether, and hit a cafe and a library on the respective days. I find that now I prefer libraries to cafes. I particularly like the enormous vaulted library study halls, where row after row of student works in silence. This is a new preference. One thing about being a grad student: you’re always taking note of your study preferences: how, where, and when do you work best? How much uninterrupted time do you need, and how many breaks? Do you need to eat before you work, or drink tea? I understand writers go through the same thing. About half of A Moveable Feast seemed taken up with this kind of detail.

So while I have generally preferred cafes and tea houses, I’m now thinking the library will be a better spot to head to. I’m thinking that the Seattle downtown library will be a good place to set up shop for a while.

I posted an algebraic question that came up in my work on fluther. I got some pretty interesting responses back, but nothing, ultimately, that will help me too much. I think the problem I was looking at doesn’t have a simple answer. I’m in a strange position anyway, because it’s easy to prove that a certain thing can’t happen, but very hard to prove (at least with what I know now) that it does happen. I need to know more. I’ve been sticking primarily to just one of the three main tools at my disposal to help me work; today I started reading more deeply into a second. I have a feeling I’ll have to have a pretty good command of all three to make real progress. There’s always this fantasy that you can learn just enough to make your breakthrough. But whenever I shy away from something, it turns out I need it later. Time to really push in to every available resource at my disposal.

Anyone who has never made a mistake has never tried anything new.

Albert Einstein