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.

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 Observation I: Algebraic Varieties

In order to define any algebraic geometry structures (a sheaf, for example), one has to first understand what an algebraic variety is. And thus:

Observation I. It’s damn-near impossible to find someone who gives a straightforward definition of an algebraic variety straight off the bat.

Instead, most authors tend to define an affine algebraic variety – first as the common zero set of a collection $\{F_i\}_{i\in I}$ of complex polynomials in $\mathbb{C}^n$ and later as a “variety that can be embedded in affine space as a Zariski-closed set” (Smith et. al., An Invitation to Algebraic Variety). Then, half a book later or more (it’s on page 144 of the aforementioned book), it’s said that an (abstract) algebraic variety is a topological space with an open cover consisting of sets homeomorphic to affine algebraic varieties which are glued together by so-called transition functions that are morphisms in the category of affine algebraic varieties.

This of course requires knowledge of category theory, the Zariski topology, etc. etc.

As of now, this 30+ minutes of searching has gotten me through about 3/4 of a page in Chris Elliott’s online manuscript concerning $D$-modules.

le sigh

Realization

Today was grocery day in the Stover household, which means we basically spent the day driving around, picking up amazing savings due to my wife’s couponing and essentially getting nothing else done whatsoever.

Fortunately, I was able to squeeze in about 30 minutes of math while sitting in the local Target’s snack bar / Starbucks area. In particular, I took some time to read a bit further into my professor’s paper on M-conformal Cliffordian functions, and in so doing, I came to a realization.

The last time I wrote here about that paper, I sketched a small proof of an elementary claim that probably required no proof. As a result of that entry, today has been as complete roller coaster for me.

First, I thought I’d misquoted the definition of a function $f:\Omega\to\mathcal{A}_n$ being monogenic: My original claim was that $D_nf=0$ for monogenic functions, but today, I miscalculated the partials for an example that led me to believe that $\overline{D_n}f=0$ was actually the criteria. That’s not correct at all.

Now, I realize what the criteria really is, but at the same time I realize that one small detail of that proof was incorrect. In particular, I combined the summed expressions for $f$ and $D_n$, respectively, to be a single sum ranging from $l=0,\ldots, n$ instead of a term for $u_0$ and $\partial/\partial x_0$, respectively, plus a sum for $l=1,\ldots,n$. Later, when there were two parameters $l,m=0,\ldots,n$ in the sum, I claimed that $l=m$ implies that $e_l^2=-1$; this, of course, is false, since $e_0$ is identified with 1 so that $e_0^2=1$. On the other hand, with a proper bit of rigor, the proof is still essentially correct.

Here’s why:

Here’s one way to think about the problem. Let $f(x)=u_0(x)+\sum_{l=1}^n u_l(x)e_l$, an equivalent representation of which is $f=\mathbf{sc}(f)+\mathbf{vec}(f)$ where $\mathbf{sc}(f)=u_0(x)$ and where $\mathbf{vec}(f)=f(x)-u_0(x)$ represent the scalar and vector parts of $f$, respectively. In particular, then, if we consider $D_nf$ to be the derivative of $f$, it follows that $D_nf=0$ precisely when $D_n[\mathbf{sc}(f)]=0$ and $D_n[\mathbf{vec}(f)]=0$. With regards to the scalar part of $f$, this implies that

$D_n(u_0(x))=0\implies\displaystyle\frac{\partial u_0}{\partial x_0}+\sum_{l=1}^n\frac{\partial u_0}{\partial x_l}=0$.

In particular, then, the vector $\left(\partial u_0/\partial x_0,\cdots,\partial u_0/\partial x_n\right)^T=0$, and so each component must be zero. Hence, $\partial u_0/\partial u_k=0$ for $k=0,\ldots,n$.

If we then turn our attention to the vector part of $f$, we see that $D_n[\mathbf{vec}(f)]=0$, i.e. that

$\begin{array}{rcl}0 & = & \displaystyle\left(\frac{\partial}{\partial x_0}+\sum_{l=1}^n e_l\frac{\partial}{\partial x_l} \right)\circ\left(\sum_{k=1}^n u_k(x)e_k\right) \\[2em] & = & \displaystyle\sum_{k=1}^n \frac{\partial u_k}{\partial x_0}e_k + \sum_{k,l=1}^n e_l\,e_k\frac{\partial u_k}{\partial x_l}\,\,\,\,\,\,\,\,\,\,(1)\end{array}$.

Note that the two sums in $(1)$ sum to zero precisely when each sum itself is equal to zero due to the linear independence of the basis elements $e_l$, $l=0,1,\ldots,n$. In particular, then, the second sum in $(1)$ equals zero and is precisely the sum I used for the matrix analogy in my original solution. Among the necessary corrections is to note that the matrices $M,M'$ cited there should be $n\times n$ matrices instead of $(n+1)\times(n+1)$. Recall also that the first equation in the system of equations shown in the original entry – the equation $\sum_{l=0}^n \partial u_l/\partial x_l=0$ – is achieved by combining the equation of “mixed partials” of the form $\sum_{k=l=1}^n \partial u_l/\partial x_l = 0$ from the second sum in $(1)$ with the fact that $\partial u_0/\partial x_0 = 0$ from above.

whew

This, sirs and madames, is what happens when one doesn’t protect against carelessness. I need to weed that out of my repertoire and fast. Blah.

Anyway, I’m gonna try to learn some Algebraic Geometry and maybe apply that to some $D$-module theory. Until next time….

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.

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….