A Confession

I have a confession to make. Unfortunately, it probably isn't a very interesting confession for anyone who isn't keen on maths. The whole thing revolves around the differences between two subbranches of mathematics which may in fact be indistinguishable to outsiders...Spend enough time in graduate school and one begins to forget that the entire world is not in fact made up of mathematicians.

But first, a headline: 'Bosnian Police Capture The Masturbator'.

Police in Bosnia say they have caught a prolific burglar who they dubbed The Masturbator.

The man had allegedly broken into scores of offices to spend hours on telephone sex lines.

'The Masturbator' sounds to me precisely like the sort of villain we'd have seen on Doctor Who had FOX produced a new series. I mean, what sort of crack-addled glue-sniffer casts Eric Roberts, as anything other than perhaps a floor lamp?

But I digress!

I've been feeling filled to the brim with a certain mathematical angst recently, not unlike my mug is now, if by 'mathematical angst' I mean 'tea'...

...It occurs to me that not all who may be reading this necessarily know me already. I should then explain. I'm a first-year Ph.D. student in mathematics at the University of Washington, in Seattle. Ever since I decided to go into maths, I've had my heart set on being an algebraist. Modern algebra bears little resemblance to the algebra we teach to schoolchildren. Modern algebra is what you get when fiendishly clever Europeans spend 150 years trying to escape from any contact with the diurnal world.

If you were lucky, and not an American under about 21, it's plausible that in high school, or your local equivalent, you were taught at some stage various properties of arithmetic, like the Associative Law, and the Distributive Law, and so forth. Algebra is what you get when you throw away numbers and take some of these laws as axioms, then study the objects that obey them. From a very small set of axioms, one gets a very large, even pervasive, class of mathematical beasts with a very rich theory describing them. The simplest sort of algebraic object is a group. A group is a set of things--what these things are doesn't matter in particular--along with a function or rule, called a binary operation, for taking any two of these things and getting a third thing from them; if we write this operation multiplicatively, as ab=c where a, b, and c are elements of our group and we mean by this that plugging a and b into our operation, in that order, yields up c (just imagine a, b, and c are ordinary rational numbers we're multiplying together), to qualify as a group three axioms must be satisfied:

• The Associative Law: (ab)c=a(bc) for all a, b, and c in the group.
• There exists some element e in our group such that for all elements a in our group, ae=ea=a. e is called an identity.
• For every a in our group there exists an element a^-1 such that aa^-1=e=a^-1a.

A simple example is the set of integers under addition, with identity 0.

But I'm digressing! The point is, as soon as I had my first abstract algebra course as an undergraduate, I knew algebra was the field for me. So I went into maths, and got into graduate school, and have been slogging away since late September in algebra, real analysis, and complex analysis.

See, here's where it gets complicated. Although I came in knowing I wanted to be an algebraist, one needs to pass a number of qualifying, preliminary examinations ('prelims') in order to become a real, honest Ph.D. candidate. Three, to be exact. And here at the UW, we have exactly five to choose from: algebra, real analysis, complex analysis, linear analysis, and manifolds. So one is obliged to learn some analysis, no matter what. Analysis, by the way, is sort of a jazzed-up version of calculus, although it covers so much territory it's hard to sum up. So I bit the bullet and went for reals and complex, even though by longstanding tradition algebraists hate analysis. (Although it's not unusual for analysts to like algebra. Everyone likes algebra.)

And here's where the trouble begins. When we got to the Lebesgue theory of integration in real analysis...I started to enjoy it. Measure theory is keen! And functional analysis has some interesting properties...Dual spaces...Now we're on Radon measures, which have some dead nice regularity properties, the sort of measures every locally-compact Hausdorff space dreams of having...And sometimes, in sick moments of twisted brain-wrongness, I think to myself, 'Gee, maybe I'd like to be an analyst.'

I even invented a piece of terminology on the current homework set. It probably won't catch on, I know, but it couldn't hurt to try. I decided that if m is a Radon measure on a locally-compact Hausdorff space X, then any open set O in X such that m(O)=0 should be called 'melancholy'. Because it's sad. Open sets shouldn't have measure zero; open sets should be big. I think 'melancholy' is a much better term for them than the other suggestion I've heard, which is 'fr-izz-eaky'. So the next time you see a null, open set as you go about your daily business, be kind to the poor thing: it's melancholy.

I live with a burning shame now, and fear; I feel hunted, marked somehow. What if my algebra chums find out I like analysis? What will they do to me? Will I become a pariah? Neither one thing nor the other? What would my parents say? What would the Church say? Can I control this impure lust? Can I defeat it, and go on to live a healthy, algebraic life? I still like algebra, after all (well, except for filthy unclean commutative rings)...Maybe I just need to represent more Lie algebras. But what if I'm weak? And Radon measures can be so, so good...Oh, the temptation. It's like living in Far From Heaven.

Only silly.

I'm sure God will smite me soon.

Posted by aloysius at April 26, 2003 12:12 PM |