# 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:

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.

# Movin’ on up (and down) (and up) (and down)….

I decided to spend as much time as possible today studying after a few days of being nonchalant with it. I went to bed early-ish last night, woke up early-ish this morning, and hit the books with very few breaks in between.

As it turns out, this recipe gave me ample opportunity to learn new things. Who woulda thunk?

I started with my professor’s paper on $M$-conformal Cliffordian mappings. I made it through a couple more pages of that guy, verifying theorems and assertions as I went along. Then, right as I was on the precipice of real math, I realized how mentally taxing my morning had been and shifted direction a bit.

My new direction: Dummit and Foote. I started section 15.2 on Radicals and Affine Varieties. About 2/3 of the way through that section, I realized I really really need to learn some stuff about Gröbner Bases, so I decided to forego that and keep the ball rolling. I spent a few minutes flipping through Osborne’s book on Homological Algebra and upon realizing I’m far too underwhelming to tackle that guy, I shifted focus again to Kobayashi and Nomizu.

Of course, K&N has kind of worn out its welcome around here, and upon reading a page or two, I decided to break out a different Differential Stuff book instead. My target? Warner’s book Foundations of Differentiable Manifolds and Lie Groups. This book is a nice amalgam of Geometry and Topology, as evidenced by its somewhat nonstandard definition of tangent vectors. Maybe I’ll share some of that later.

Finally, I decided to shift my focus back towards Algebraic Geometry, whereby I broke out Eisenbud and Harris’s book The Geometry of Schemes and tried to stay afloat. Much to my own surprise, I was able to make it through fifteen-or-so pages without floundering completely and/or ripping all my hair out, so I’m hoping that maybe the information I’ve picked up in other places has done me some good. We’ll see for sure moving on.

Overall, I think I cranked out about 45-50 pages of reading today – and all (well, most) on material that’s completely new. It ain’t a Fields Medal, but it ain’t a flop either.

Until next time….

# Differential Geometry, or Frustrated Beyond Frustration

So yesterday, I came here and vented about how I was trying to prove a trivial mundane fact from Kobayashi and Nomizu and how – despite struggling for hours – I felt close but still rather hopeless.

Enter today.

My game plan originally had been to spend today doing some of the studies I’d been neglecting recently (namely, Clifford analysis and Differential Algebra) but I really didn’t want to leave that identity without a proof. Then, I hashed a new plan: One hour on that identity, then split the day between Clifford analysis and Differential Algebra. Pretty simple, right?

Wrong.

After spending some time running morning errands, I sat down with the identity and went to work with some new strategies. Instead of trying to argue a notation-heavy thing from an abstract existence point of view, I reached into my manifold $M$, picked out a point $p$ with neighborhood $U$ and gave $U$ a local coordinate system $u_1,\ldots, u_n$. Then, I went to town, hitting shit with Vector Fields and composition and compositions of Vector Fields of compositions of vector-valued functions of….

…and now, here I am. It’s been two hours (and some change) since I started that identity and still, no proof whatsoever. So that got me to thinking: What if I really don’t understand properties of vector fields at all? What if I’m not a differential geometer or a differential topologist or a differential algebraist? What if I’m a differential nothing-er? What if my career is over before it starts?

So I decided to turn to Google.

I did some random search involving the terms “vector fields” and “composition” and – lo and behold – I found something that’s actually meaningful. And – and – I realized what my problem is…

I’m using a book published in 1963. It’s a classic. My book is a first edition of a text that’s so classic its reprints are selling on Amazon for almost \$200. It’s a gem. But 1963 was 50 years ago.

Fifty. Years. Ago.

My book is archaic and reading it is getting me into trouble.

I know this because my Googling turned up this wonderful resource, and upon reading lecture seven, my answer was immediately clear.

Basically, I was attempting to prove the Pullback-Pushforward identity (given here in K&N notation): Given a manifold $M$, a diffeomorphism $\varphi:M\to M$, and a vector field $X\in \mathbf{X}(M)$, along with points $p,q\in M$ for which $p=\varphi(q)$, the induced homeomorphisms $\varphi^*:\mathbf{F}(M)\to \mathbf{F}(M)$ for which $(\varphi^*f)(p)=(f(\varphi(p))$ and $\varphi_*:\mathbf{X}(M)\to\mathbf{X}(M)$ for which $(\varphi_*X)(p)=(\varphi_*)_q(X_q)$ interact in the following way:

$\varphi^*\left((\varphi_*X)f\right)=X(\varphi^*f)$ for all $X\in\mathbf{X}(M)$ and all $f\in\mathbf{F}(M)$.

Simple, right? Apparently the relationship I was missing is given in equations $(7.2)$ and $(7.3)$ in the seventh lecture linked above.

So now I have a dilemma.

I need to study Differential Geometry this summer. I’ve already invested almost a week – maybe a little more – in going through 10 pages of this book and trying to hash out the details. Now, I realize that with the archaic notation, 50 pages by summer’s end would be a good pace for someone as average as myself and how that’s just not good enough.

Clearly, newer documents are easier to read: The notation is cleaner and more intuitive and the exposition is aimed more at educating readers than simply expounding upon one’s own knowledge of the subject. Clearly I would have better luck progressing through documents like that.

Clearly, those are better for me.

But then it feels like a loss somehow, ya know? Like I started at the bottom of a mountain, scratched and clawed my way up through arduous paths and unfriendly conditions, and have come to a point where someone’s waiting in a helicopter to take me back up the hill to safety…

…and okay, then: My metaphor just answered my own question!

I guess tomorrow – or Saturday – or whenever – I’ll start studying Differential Geometry using some other textbook not written in the 60s (which means I’ll also probably skip out on Auslander’s book, too).

Until next time….

# Differential Geometry Sunday

I was able to actually stick with my plan a little earlier and spend the day parsing through some stuff in Kobayashi and Nomizu. That’s not a bad way to spend a Sunday.

Somewhere in the middle of that, I ended up stumbling upon something I’d always been somewhat privy to and I did so almost by accident. In the text, I ran across the definition $f_*$ in the following context:

Consider two manifolds $M$ and $M'$ and a mapping $f:M\to M'$ of the prior into the latter. Then for a point $p\in M$, the differential of $f$ at $p$ is a linear mapping $f_*:T_p(M)\to T_{f(p)}(M')$ which is defined as follows: Given a vector $X\in T_p(M)$, choose a path $x(t)$ with $p=x(t_0)$. Then $f_*(X)$ is the vector tangent to the curve $f(x(t))$ at $f(p)=f(x(t_0))$.

The notation reminded me of something I saw during my very first foray into Differential Geometry, namely the Hodge star/dual operator. It was a notion that was so novel when I first saw it that I contemplated preparing a seminar talk at BGSU for my peers, none of whom were geometers of any kind; now that I’ve rediscovered it, I’m having similar ideas for my non-geometer peers here at FSU. But I’m getting ahead of myself…

# Working leisurely or Doing nothing?

So here’s the thing: I haven’t really done anything today. What I mean is that I haven’t constructed anything new (a page, a list of definitions, a solution) that didn’t exist yesterday, and so – for all intents and purposes – I haven’t done anything.

But somehow, I haven’t done nothing either.

Some days, I make a plan to do something (“do” something), and I set out on that path. Sometimes, the path I reach has a bunch of hurdles that I’m not prepared to conquer, and so I set out on a side journey to obtain the skills necessary to progress down my original path. Sometimes – on days that are particularly unkind – the side paths have hurdles requiring sidepaths and the side-side-paths have hurdles requiring side-side-paths and so the whole journey gets twisted into some amalgamated blob of non-progress that somehow still manages to accomplish something.

That, ladies and gentlemen, was a metaphor. It’s a metaphor that fits my day rather well.

So as I mentioned earlier, algebraic topology was a bust. I decided, then, to finally take the plunge and to read something on $D$-modules via Google. My professor had suggested this as a nice algebraic way to derive lots of the differential geometry results by way of learning really difficult algebra stuff like categories and stacks and sheaves and schemes and what not. That, of course, got me interested. I did a little digging and found an online resource from Harvard and decided to take a stab. I made it through about a page before I realized I was missing stuff on stuff.

I freshened up on stuff about Lie groups and took a gander at what Wikipedia had to say about Universal Enveloping Algebras. Of course, Universal Enveloping Algebras required me to know things about Tensor Algebras, and when I decided to look up something more foundational like “rings of differential operators”, I decided that I should probably concurrently try to parse through some literature regarding Differential Algebra as well. That chase has brought me to where I am now and has sustained me for the better part of three hours.

In that three hours, I’ve found lots of good resources (including an online .pdf of Ritt’s text Differential Algebra) and have done quite a bit of reading, but if I were to die today and pass the totality of today’s efforts off to someone else, their inheritance would consist of precisely zero tangible work.

So yea: Not doing anything while not doing nothing is a thing and it’s called “research mathematics”. Such is life, I suppose.

I think I’m going to end today’s part of my quest on the differential algebra / $D$-modules front here: I’ve got some stuff to do and what not, yadda yadda yadda, etc. etc. I plan on working some more on Kobayashi and Nomizu before bed, though.

All I do is math math math no matter what….

# Differential stuff

Well, I finally did it.

I finally pried myself away from the overwhelming constancy that was Allen Hatcher’s book and began devoting some time to differential stuff. Don’t believe me? I even made pages full of differential stuff! See for yourself!

Anyway, I’m currently sifting through the terse mathematical jungle that is Kobayashi and Nomizu. I have other resources at my disposal (perhaps more than I can admit without feeling guilty of hoarding mathematics), but I wanted to give this a try since this original two volume set was a gift to me by Dr. So-Hsiang Chou, an applied mathematician who coincidentally is probably one of my all-time favorite academics I’ve met thus far.

Dr. Chou was full of valuable insight about his main research areas, the main research areas of pretty much anyone else, and just about anything you could possibly want to know about life. Sincerely, I love that guy. He had lots to say about my passion for differential stuff. He told me it was too hard and that I should find something easier to do my thesis on; he told me it was going to make me go insane; he also said, very objectively, that it’s extremely notation-heavy and that any resource less than an intro to an intro would be a hellish endeavor on just about any brain.

He was certainly right about the latter.

Anyway, at this pace, I’ll make it through Volume 1 of K & N by the time I’m 3,000 years old, so I plan on supplementing my perusing via perusing of other sources too; as that materializes, I’ll probably use the above-linked page to help there, as well.

That’s enough of a break for me. I’ve read through about 9 pages of K & N today – teX’d the high points of about five of them so far – and I definitely need to do more before calling it a night.