Study plans, or Why it’s embarrassingly late into the summer and I still haven’t finalized a good way to learn mathematics

So it’s now creeping into the third (full) week of June. School got out for me during the first (full) week of May. Regardless of how woeful you may consider your abilities in mathematics, I’m sure you can deduce something very clear from these facts:

Summer is about half over.

Generally, that fact in and of itself wouldn’t be too terrible. I mean, big deal: Half the summer’s over, and I’ve been working throughout. How big of a failure can that really be?

In this case, it’s actually a pretty big one.

Despite my having read pretty much nonstop since summer began, I haven’t really made it very far into anything substantial. Compounded onto that is the fact that I’ve had to abandon a handful of reading projects after making what appeared to be pretty not-terrible progress into them because of various hindrances (usually, a lack of requisite background knowledge).

It’s been a pretty frustrating, pretty not successful summer, objectively.

Among my recent failures in differential geometry are my oft-written-about issues with Kobayashi and Nomizu, which led of course to beginning to look at books by Auslander, Wells, Warner, and Lee (this one and this one). I never really gave Auslander a fair shake before that one ended up on the bottom of the stack; Wells’ book is disappointing because the later material is precisely what I’d be looking for in terms of combining machinery from different subdisciplines, but by trying to generalize the behavior of differential, real-analytic, and complex-analytic manifolds, the notation is clumsy and unintuitive. I like Warner overall but the exercises seem pretty well beyond the scope of what the text addresses, and, by and large, the first of Lee’s books listed there is good but is more topology and less geometry. The second of his books is more geometry, but after dredging through 40-ish pages in a day, I hit new material with the realization that I didn’t know the requisite material very well.

Today I decided once and for all to suck it up and start working through Spivak’s big book(s). I cranked out about 40 pages there today – even making progress on some exercises – before calling it quits. My summer-long suspicion that I’d be behooved to start working through his small book as well was confirmed, so I think those two + Warner will constitute my differential geometry plan moving forward.

Then, there’s algebra. I need to work on the commutative algebra book; I need to learn algebraic geometry. I know I need to focus on Eisenbud/Harris and on related resources needed to acquire that knowledge successfully, but I can’t seem to shake the voice in my head that tells me to focus on Dummit and Foote first. That uncertainty is furthered by the lack of specifics received from faculty members I’ve emailed. Oh well.

I also haven’t been able to hash out a clear avenue of communication with the professor whose Clifford analysis paper is now an increasingly-sparse, overwhelmingly-frustrating and difficult part of my routine. There really is no excuse for me being as slow as I have been on that front, and the more I progress, the more I realize I’m being forever shitted on by my having never taken a graduate-level linear algebra course. I have a rather highly-regarded text downloaded for that purpose; perhaps that should also somehow find its way into my routine.

Guess I need to make zero hours become not zero hours. There’s a thesis topic for me to ponder.

Somehow, too, I’ve not mentioned Hatcher or algebraic topology in general, nor have I begun to fret about my summer Foundations of Mathematics class which begins one week from today.

The more I stare at this screen and keyboard, the more depressing it becomes.

So that’s it…first things first moving forward will be to devise a study plan. That’ll be the beginning of tomorrow for sure.

Until next time, au revoir!

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