# 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….

# CW-complexes

So as I spend my days progressing through the very dense, very slow-moving genius that is Hatcher’s book Algebraic Topology, I’m constantly reminded of things I’m not very good at.

And believe me: There are lots of things in that book I”m not very good at.

One thing I’ve always struggled with were the technical details of CW-complexes. I’ve mentioned that before. As a result, I’ve spent the better part of an afternoon gathering online resources, etc., that would shed insight onto the things I’m not clear about. While I wouldn’t bet money on this, I feel confident that I’m now better-equipped to recognize and understand the theoretical construction and properties of CW-complexes…

…and yet, I still find that I can’t do many (as in, I can do almost none) of the problems in Hatcher.

I’m pretty sure I understand the gist for problem 0.14 (for example), which asks you to put a CW-cell structure onto $S^2$ with $v$ 0-cells, $e$ 1-cells and $f$ 2-cells, where $v,e,f$ are integers for which $v-e+f=2$. The gist seems simple: Add on a cell in a precise, regimented way, and define a characteristic map which correctly “incorporates” the additional construction into the given construction. I get that. But where do I start?

I just don’t seem to know enough to transition from sticking my toes in the water to jumping in and taking a swim. I’m pretty sure jumping in at this juncture means drowning.

As I’m sometimes inclined to do, I dug up some other solutions to see if they could shed insight. I feel like Tarun’s solution is overly complicated, while I feel like Dr. Robbin’s solution assumes way way more knowledge than I have at my disposal.

This leaves me feeling a little lost on the algebraic topology front, meaning I’m going to have to dig through some of the resources I have at home and try to figure something out.

Gah.

For completeness, it should be noted that I’ve scrounged up a couple of resources online: Find them here and here.

Maybe in the meantime, I’ll dig up a diversion or two: I’m thinking some differential geometry or maybe even some $D$-module theory. Maybe I’ll be adding some more around these parts later.

In the mean time, check out these easy-to-read, casual expositions on the proofs of the ABC conjecture and the Bounded Gap Conjecture for Primes. I find the first story particularly intriguing.