Week 3, Day 1 or Properties of Lie Brackets

Today is the first day of the third week of the semester. I know that counting down like this is going to make it seem longer than it already seems, but it seems so long that I can’t seem to help remaining conscious of the precise time frame I’m dealing with.

Such is life, I suppose.

I’ve noticed an amusing trend in my page views involving solutions from Dr. Hatcher’s book, namely that I’ve been receiving an abnormally-high level of page views lately, almost all of which seem to center on those solutions. I guess that means that the semester has started elsewhere too and that people find topology as difficult and frustrating as I do.

For those of you who fit this bill and who are reading this right now: My plan is to start doing more problems ASAP, so that page might get its first update in quite a while.

Today, though, I want (read: need) to talk about differential geometry. In particular, we spent some time in class last week discussing the Lie bracket and its properties, and because we have a derivation of one particular property, I wanted to take the time to put that here for my own benefit.

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Some Exercises from Lee’s Introduction to Smooth Manifolds

These are exercises from the first few pages of Lee’s Introduction to Smooth Manifolds. These are pretty elementary from a topology perspective – maybe the middle-to-late part of a semester on point-set topology – but I decided to put them here to (a) remember some of that stuff, and (b) force myself to not delay doing them any longer. As it turns out, I struggled more with these than I should have and even had to consult the internet for a couple thing; in instances of internet borrowing, I provide links.

Note, finally, that if you’re trying to hunt these particular exercises down within Dr. Lee’s book, they’re actually dispersed throughout the text; suffice it to say, I’m not a fan of that particular method of delivery. Nevertheless….

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Yesterday, Today, and Forever

Yesterday was a day filled with reading.

Also, by and large, yesterday was a day consisting entirely of (differential) geometry / topology, so it’s really no surprise that – again – my dreams were all math related and tied to that general realm of theory. More precisely, I spent my entire sleep cycle pondering the Poincaré Conjecture (can we call it the Perelman Theorem yet?) and Ricci Flows. That’s certainly a night well spent.

Unsurprisingly, my day today will be largely similar. I downloaded a bunch of resources concerning the aforementioned topics (Poincaré-Perelman and Ricci Flows), as well as some (more) texts on Riemanninan Geometry (which I started perusing yesterday). Also in the works: A colleague of mine (who I’ll call DW2) and I have decided to work through Atiyah and MacDonald’s Introduction to Commutative Algebra, and I’m pretty sure if I don’t spend a significantly-larger amount of time on my professor’s Clifford paper, I’m going to have zero things about which to ever talk with him…

…then there’s the algebraic geometry stuff I’m working on in Eisenbud and Harris / Dummit and Foote, and the material from the seven or so other books I’m reading through concurrently right now….

Every day I’m huss-uh-lin’….

I have some things I want to write here later – expository things and what not – but for now, it’s just this check-in. Auf Wiedersehen!

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….

de Rahm Complexes: Really cool math, or a miracle?

Before explaining the title, here’s a little background:

Let \Omega^* be the vector space over \mathbb{R} with basis

1,dx_i,dx_idx_j,dx_idx_jdx_k,\ldots,dx_l\ldots dx_n.

Using this notation, the collection \Omega^*(\mathbb{R}^n) of C^\infty differential forms on \mathbb{R}^n are elements

\begin{array}{rcl}\Omega^*(\mathbb{R}^n) & = & \left\{C^\infty\text{ functions on }\mathbb{R}^n\right\}\otimes_{\mathbb{R}}\Omega^*\\[0.5em] & = & \oplus_{q=0}^n \Omega^q(\mathbb{R}^n)\end{array}

where \Omega^q(\mathbb{R}^n) consists of C^\infty q-forms on \mathbb{R}^n. There also exists a differential operator d:\Omega^q(\mathbb{R}^n)\to\Omega^{q+1}(\mathbb{R}^n) which satisfies the expected properties for exterior differentation. Under this construction, the pair \left(\Omega^*(\mathbb{R}^n),d\right) is called the de Rahm complex on \mathbb{R}^n. Moreover, the kernel and image of d are known as the closed and exact forms, respectively.

When this material was presented in Bott & Tu’s Differential Forms in Algebraic Topology, the following quote was included:

The de Rahm complex may be viewed as a God-given set of differential equations, whose solutions are the closed forms. For instance, finding a closed 1-form f\,dx+g\,dy on \mathbb{R}^2 is tantamount to solving the differential equation \partial g/\partial x-\partial f/\partial y=0….

So maybe there is a God, and maybe God is a mathematician? *ponders*

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….