Algebraic Geometry, or Why I’m a moron

There are two main things to note before I hit the main components of this rant tryst exposition. Number one:

In my department, Algebraic Geometry is a big deal. We have two (count it: one, two) algebraists whose expertise in the subject is second-to-none, and we have another cluster who – despite being cross denominational in their research – are truly masters in the field.

In my department, Algebraic Geometry is a big deal.

It’s unsurprising, too, I guess: Since Grothendieck revamped the field in the 50s and 60s, its usefulness has been realized to be extremely wide-spread and, as such, people really really care about it.

In my department, Algebraic Geometry is a big deal. That’s the first thing to keep in mind.

The second thing to keep in mind is that I’m a very ignorant person. I tend to form uneducated judgments under the premises that they’re not at all uneducated, only to find out later that my perception was wrong, that my brain was nothing but ignorant, and that afterwards – invariably – in some way – I’m going to have to suffer as I attempt to either (a) catch up, or (b) give up and mourn.

Now what’s the point of this entry?

Several of my colleagues here are blossoming algebraic geometers: They’re people who recognized the importance of the field (while happening to be good at its constituent components), who identified our institution as one worthwhile in terms of like-minded research fellows, and took the time to uproot their lives to come and be a part of the cutting edge. I happen to be good friends with a couple such people and I can tell you that both are very apt – very very apt, to the point I’m envious most times – and that they’re both very passionate about the research that’s waiting for them in the not-so-distant future.

These two guys aren’t ignorant.

I, on the other hand, came here knowing absolutely zero things about algebraic geometry. In fact, my first exposure to those two words meshed as a phrase was when I was applying to Ph.D. programs: It was then that I discovered that lots of people liked and did and succeeded in research concerning algebraic geometry. Being in my own little world, though, I kept my small-sighted, narrow-minded deduction of areas I knew and liked and wanted to pursue and never bothered learning what all the fuss was about. Then, even when I got here, one of the aforementioned couple took the time to tell me about what he was studying: I knew nothing about it, agreed to participate in an ad hoc reading group he put together, and ended up never being able to attend a single meeting.

So again: I was, am, and continually have been out of the loop.

In this case, though, it’s even deeper than that because in this case, I was truly not interested in the (sub-)area he talked about. One of the things he was most interested in was so-called intersection theory which – as the name suggests – hinges on the study of properties and behavior of intersections of two “curves”. He told me about this and about how his potential advisor was a super-hotshot-rockstar-math-god at this study and as he was talking, I decided that I was really, truly not interested.

And this, folks, is a big deal. You see, in my entire mathematical career (I’m an old man, remember), I’d never once not liked an area of math. I liked some more that others, but I always felt like I’d consider myself a failure for not being able to master all of the areas fully: To me, mathematics is everything and anything less than complete mastery of it means I will have let myself down as a hoarder of mathematical knowledge. Literally, I wanted to know all the math…

…except algebraic geometry…

…and now, here we are.

Since last writing about the analysis of Cliffordian functions, I decided to spend some time researching what I’d been calling Differential Algbera, a phrase I picked up from Ritt’s book of the same name and one that I was (rather incorrectly) throwing around to describe the objects I really wanted to study – namely, rings of differential forms and the behavior of modules on said rings. These are the so-called D-modules. After reading some of Ritt’s book, deciding it was irrelevant to what I cared about (sorry, Dr. Ritt), and digging up the two manuscripts I’d downloaded specifically aimed towards D-modules, one thing became abundantly clear:

The amount of algebraic background I need to make up before understanding even something basic is probably about the same as the amount of algebraic knowledge I currently possess.

That’s disheartening.

Disheartening, too, are the objects of which I lack knowledge – the objects whose names come up nearly every sentence in any of the manuscripts I found:

Schemes. Stacks. Sheaves of differential forms. Affine varieties.

As sure as I’m writing this entry right now, I can attest that the thing I’m lacking most soundly with regards to being able to study D-modules is precisely the field I’d shrugged off ignorantly fewer than two semesters ago….

Ladies and gentlemen: I need to become an (amateur) algebraic geometer.

dims the lights

Admittedly, I panicked a little. Having seen the depth that this field possesses and the amount of effort functional understanding of it requires made me realize that the time to wait ended about five years ago: I have to make progress and fast. One way I started on this path was to accumulate resources: My “Differential Algebra” folder that once housed three lonely, oft-overlooked documents now contains 16 documents of various sizes and depth. I’ve downloaded a few undergraduate-level introductions, and a couple of documents at each of the logical stages in between.

It’s obvious now what I have to do, and so do it I shall…

…starting tomorrow, probably, when I’m not as exhausted, mentally, as I seem to be right now….

So if you guys don’t see me around for the next ten or twelve years, you know I’m a recluse, living dirty and homeless outside the bookstore of some random top-10 research University and mumbling babble about the category of sheaves over a particular topological space along with their morphisms. You guys can thank Dr. Izzo and one of his countless wonderful, fascinating stories for this image.

I guess that just about does it. Until next time,….

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