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

Good Mistakes, Constant Mistakes

I’ve spent the last two days going over my problem, going over my approach, finding new gaps in my proof, fixing them… wash, rinse, repeat. It’s amazing that this vision of math as “getting to the right answer on your first try” even exists. I have to make, unmake, remake so many mistakes to get where I’m going. I think all mathematicians work that way.

Einstein famously said, “Do not worry about your problems with mathematics, I assure you mine are far greater.” I don’t think he said this to brag about how advanced he was: I think he genuinely had a lot of trouble with math. Somehow, a big part of the experience of math is trouble. Frustration is the status quo. But when you get something—the thrill!

Speaking of which, I’m feeling pretty good that I seem to have my argument at least mostly patched up at this point. I think I benefited quite a bit from certain mistakes I made along the way. To quote Simon Singh’s Fermat’s Enigma:

While Shimura was fastidious, Taniyama was sloppy to the point of laziness. Surpisingly this was a trait that Shimura admired: ‘He was gifted with the special capability of making many mistakes, mostly in the right direction. I envied him for this and tried in vain to imitate him, but found it quite difficult to make good mistakes.’

Math doesn’t happen in a straight line. If I hadn’t made as many mistakes in my thinking about this problem, I don’t think I would have solved it.

Of course, I should sleep on it to see if I have a real solution. Can’t sound the fanfare too quickly.

So Close–Update from MSRI

I’m attending a conference in Berkeley, CA right now, at the Mathematical Sciences Research Institute, lovingly dubbed MSRI (pronounced “misery”). When I first started attending conferences I felt good if I understood the first half hour of a one hour talk; that was a very rare occurence in the beginning. It was common for the speaker to lose you in the first 5-15 minutes. Maybe that’s why conferences were so tiring—seems like I spent most of the time groping to understand.

Now that I’m (somewhat) more mature as a mathematician, I’m noticing two things. First, the talks that are in my area are comprehensible. It’s deeply gratifying to know what’s going on. Second, when a professor lectures on an area that I don’t understand and am not interested in, I have no qualms about ignoring him and doing my own work. I would estimate that two thirds of a typical audience at a math conference isn’t paying attention to the front of the room.

More exciting than the talks were the fact that I got engaged in a problem I’ve been working out: a new thought on how to proceed with it and a pair of good papers to adapt to my own needs, and suddenly I’m putting all my time into making it work. I thought I solved a subproblem yesterday, and went so far as to write it up and email it to my advisor. When I revisited it the next morning, though, I caught a mistake.

So close…

Now I’m trying to see if I can make the thing work. It’s productive work, even if it doesn’t work out. I’m starting to understand how (technical gibberish alert) the K3 surfaces relate to their lattices and to their Fourier-Mukai partners (end alert). I think I may have figured out how to close the gap in the proof, but I’m a little hesitant to declare victory. In fact, I’m thinking there may be a few more wrinkles ahead. Hopefully it will come through.

So far…

In the meantime, though, I’ve got plenty to do when the speakers veer into topics that I don’t care for.