Saturday, May 5, 2007

How I learned algebraic topology

Some people can learn out of books. This is actually a good option if you want to learn algebraic topology nowadays; there's plenty of good books, some even in my native language, and some can be downloaded for free (legally!).
But I'm not good at studying on my own, so I took a course in my second undergraduate year. The course was attended by a grand total of two students, and none of my friends took it. It turned out to be a fabulous course. The professor, undaunted by the paucity of the audience, gave a series of beautiful lectures, covered a lot of material some of which wasn't even published yet, and gave us a huge amount of handwritten notes to support the reference textbooks.

Algebraic topology is a strange beast. In a sense, it is extremely intuitive: it gives precise meaning to statements like "a donut has one hole, but a pretzel has three" and allows you to generalize such statements to objects you can't visualize at all. However, when you seriously want to study it, you discover you need a lot of technical tools, and it's easy to loose the intuitive aspects. The handwritten notes we were given were largely hand drawn; it's thanks to them that I kept my intuition alive all that year, and a combination of that and of differential topology has become a substantial part of the subtly warped bag of mathematical tools that allow me sometimes to see more, or differently, than many more talented and informed colleagues.

Most of all, I was fascinated by the professor herself. She kept her hair short, clean but untidy, wore baggy jeans and a poncho, and didn't try to hide either the first few gray hair or a reasonably substantial moustache. She was tall, strong, loud, and wore thick nonfashionable glasses. She brought to the course an incredible enthousiasm, and brought us so far that we each decided to read a piece of more advanced stuff on our own instead of the usual question-and-answers exam. In my case, I studied a whole book. I didn't understand it very well, and the professor was not terribly impressed. She still gave me a top mark, although when she saw that I had top marks in all the mathematics exams, she jokingly regretted having done so.

The following summer, I sat down with a helpful, older student to figure out whom I should choose as thesis advisor. I knew: I wanted her, the woman who looked like me, the person who could bring homological algebra to life. Plus, she was working on an exciting project for a concrete description of something exotic called cohomology. But my friend discouraged me: he said that I should choose instead my future advisor, who was a rising star in a field which is popular in my country, while algebraic topology professors are few and far apart. Plus, he added, the professor I so liked had been promoted and was leaving anyway.

The advice was completely correct. My advisor and I have been a fantastic match, and the algebraic topology professor left mathematical research for didactic of mathematics; she also never came back to my alma mater, and I have seen her once only since, about ten years ago. As for the situation in my country, in each of the three universities where I have worked so far I have been the algebraic topology reference person, based on that one lonely course so many years ago, because most people didn't even know as much.

Besides teaching me a fantastic amount of mathematics, she taught me that I didn't have to look like my mother told me all women should look like; that I could choose to wear comfortable rational clothes, and derive intense pleasure from a job where nobody cared how I looked like. Even more importantly, she taught me that it is not important how many people are listening to you: what counts is that you always do your best.

I didn't understand all of this immediately, and I would certainly not have been able to put it into words at the time. I kept wearing makeup, and skirts, and uncomfortable shoes. But the she has helped me become the myself that I am today. And my first paper with WS has two main theorems, and not one, because I knew enough hands-on algebraic topology (who knows, maybe it even helped me impress WS?).

I wouldn't say my Algebraic Topology professor ever felt like a mother to me. She actually didn't look motherly at all, despite a very feminine body that the poncho disguised but became pretty obvious with the arrival of summer. But when I read that next month's topic is mothers and others, women who have influenced us along the way, I immediately thought of her.

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