# Doors closing, opening

So it’s been a hectic few days around these parts, in part because of things happening on the work front and in part because tomorrow is the first day back to school for me after a six week hiatus. It’s bittersweet, really.

By and large, the learning part of school makes me happy; I guess that’s a given since it’s a career thing for me, now. Tethered to that aspect are the things that are less-pleasant, among which are miscellaneous other duties, etc. I’ll be taking one class which, for all intents and purposes, seems like it’s going to be amazing; I’ll also be spending around 8 hours per week doing TA duties, and trying to split the remainder of my time between continuing the work I’ve been doing throughout the summer, balancing work-at-home things, and seeing about an internship that may be beginning soon.

Lots of things to keep me busy; I’m not sure I’ll necessarily be enjoying it all, though.

In other news:

I spent today being mostly idle on the math front. My plan was to have a carryover of yesterday’s supposed Algebra day since yesterday was spent mostly idle on the math front, as well. Today consisted of lots of not feeling well, running errands, and sleeping randomly. After all that subsided, I tried to work some of the exercises in Eisenbud and Harris only to be re-re-re-reminded of how important it’s going to be for me to get a good book that incorporates category theoretic ideas into some kinds of examples so I can see how to use ideas instead of just read them.

Seriously, though: I’ve read the handful of equivalent definitions of direct limits about 300,000 times, and I’ve scoured the internet to see how people respond to other people asking how to compute them, and still: I have no idea what I’m actually trying to do. I’m not sure how many times someone has to read and reread the same four pages on sheaf theory before something clicks, but I’m starting to grow anxious.

Maybe I need to start looking in other resources.

Besides that, I’ve got nothing: Failed attempts at Eisenbud/Harris solutions and lots of time spent being unmotivated. Nicht gut.

Good night, everybody.

# Late nights and early mornings

I woke up 16 hours ago and spent almost every minute of the day juggling algebra stuff: I spent a bunch of time alternating between $D$-modules and the algebraic geometry preliminaries I needed to understand that. Then, around 6:30pm, my brain sort of…went to sleep.

I took a break for dinner and decided I couldn’t just waste my time before bed, so I decided to spend some time solving some of Dr. Hatcher’s problems. I posted a couple new solutions here and here.

Now, it’s almost 1am. Unsurprisingly, I feel like I’ve gotten a second wind, so maybe I’ll try to do some more reading, or some more sorting through professors’ research, or some more algebraic topology problems, or some more….

Good night, everyone.

# Algebraic Geometry Observation II: Sheaf Theory

Observation II. Sheaf theory is hard.

Per my earlier entry: Given a smooth complex algebraic variety $X$, I finally manage to track down a semi-manageable definition for the structure sheaf $\mathcal{O}_X$. But here’s the thing with super-abstract definitions of things:

There’s a big difference between “getting them” and understanding them. In this case, I read the definition a few dozen times, annotated the .pdf file by Elliott with what I found, and felt as if I truly “got it”. Then, I read an example regarding the topological space $X=\mathbb{C}^n$ with the claim that $\mathcal{O}_X=\mathbb{C}[x_1,x_2,\ldots,x_n]$

…say what now?!

Finally, I reread a bunch of stuff and found a cool analogy between arbitrary varieties $X$ and the case where $X=M$ is a smooth $k$-times continuously differentiable manifold of dimension $\dim X=n$. This cleared things up for me.

It really is going to be a long summer. Heh.

# 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.