Another Sunday, or Awaiting Week 4

3 weeks.

I’ve officially survived the first three weeks of my second year of grad school (twice, actually). Again, I know keeping count of the days is a terrible thing to do to myself, particularly when there’s been a very small amount of good to come from my weeks thus far, but at this point, I’m sort of using that countdown as some sort of badge of accomplishment. Or something.

The coming weeks are going to be very very stressful and busy and stressful. Besides my usual load of stuff (I’m enrolled in 6 classes, I have a reading class in algebraic geometry starting up on Tuesday, I’m TAing for 1 lecture and 7 labs, and I’m trying to pick advisors / plan presentations I’ll need to give some timem soon), I also thought it was a good idea (remind me why?!) to make a poster to present at FSU’s upcoming Math Fun Day. That particular endeavor shouldn’t be especially difficult, but it requires time and time, ladies and gentlemen, is precisely what I have zero of.

Daunting is the adjective that comes to mind.

Also daunting is / was / has been the thought of continuing my goal to do all the problems in Hatcher. As you may recall, I spent the first half of summer slaving to acquire the information needed for the Chapter 0 exercises, only to have my plan for Chapter 1 totality derailed by that little piece of awesome that was my Wolfram internship. Long story short: The obsessive-compulsive part of me wants to not move forward until I hash out a Chapter 1 plan, but the This will benefit me in the class I’m taking now which, subsequently, hinges on my ability to understand Chapters 2 and 3 of Hatcher part wants to press forward.

I’m pleased to announce that the second guy won out.

In particular, my Hatcher Solutions page is showing signs of progress. It didn’t take as long as I’d predicted it to take to build that framework, and due to a random, unforeseen bout of sleeplessness at 3am this morning, I had precisely the opportunity needed to seize the moment. Right now, all those are empty pages, but I’m pleased to report that I seem to have accumulated approximately six solutions; if everything goes as planned, I’ll be taking time to update by including those as soon as possible.

In the meantime, I’m going to continue to hash out what to do about this paper. And what to do about the professors I’m emailing regarding potential advisor-hood. And what to do about the fact that I severely cut my weekend work time by spending yesterday ballin’ out of control in celebration of my wife’s birth. And what to do about….

Au revoir, internet. I bid thee well.

Oh, I just remembered: I have my first exam of the semester Friday. It’s on field theory. I’m less than pleased. :\

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.

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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!

The Half-Week That Never Was

As I type this, it’s 2:45am on a Wednesday. I haven’t been around these parts since Sunday night (actually, 3:30am Monday morning), so one would think I’d have accumulated a ginormous list of professional doings to post proudly about here.

I regret to inform: That is not the case.

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Sunday Summary

My mathematizing wasn’t very impressive today. I:

  1. Read some pages on Gröbner bases in Dummit and Foote.
  2. Did some tutoring / tutor-related things.
  3. Spent some time figuring out some solutions from Hatcher.

Despite it being 3:30am, my game plan is to be up around 8am to make a trip to campus. While there, I plan to do the usual errand things, and to then lock myself in my office for 5-or-so hours and do some legit math things.

That means I need to take good resources there with me.


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….

Verifying Easy Properties, or Nowhere, Going Nowhere

Whenever I decide to learn something – and especially when it’s learning for learning’s sake – I make sure to be meticulous with things. In particular, whenever I see propositions stated without proof, I break out the old pen and paper and start verifying.

The purpose of this post is to examine a few of the properties on page 661 of Dummit and Foote. Some of the background notation needed is discussed in this previous entry.

Claim. The following properties of the map \mathcal{I} are very easy exercises. Let A and B be subsets of \mathbb{A}^n.
   (7) \mathcal{I}(A\cup B)=\mathcal{I}(A)\cap\mathcal{I}(B).
   (9) If A is any subset of \mathbb{A}^n, then A\subseteq\mathcal{Z}(\mathcal{I}(A)), and if I is any ideal, then I\subseteq\mathcal{I}(\mathcal{Z}(I)).
   (10) If V=\mathcal{Z}(I) is an affine algebraic set then V=\mathcal{Z}(\mathcal{I}(V)), and if I=\mathcal{I}(A) then \mathcal{I}(\mathcal{Z}(I))=I, i.e. \mathcal{Z}(\mathcal{I}(\mathcal{Z}(I)))=\mathcal{Z}(I) and \mathcal{I}(\mathcal{Z}(\mathcal{I}(A)))=\mathcal{I}(A).

Proof. (7) Note that A\cup B= (A\setminus B)\sqcup (B\setminus A)\sqcup (A\cap B). In particular, a polynomial f vanishes on A\cup B if and only if it vanishes on each disjoint component of (A\setminus B)\sqcup (B\setminus A)\sqcup (A\cap B) separately, which happens if and only if it vanishes on the entirety of A and on the entirety of B.

(9) Suppose A\subset\mathbb{A}^n. Clearly, any polynomial f which vanishes on A is in \mathcal{I}(A), and because f\in\mathcal{I}(A) if and only if f\equiv 0 on A, A is certainly contained in the zero set of f. Therefore, A\subseteq\mathcal{Z}(\mathcal{I}(A)). Note that the inclusion doesn’t necessarily reverse since f(x_1,\ldots,x_n) may vanish for some element (x_1,\ldots,x_n)\not\in A.

Next, suppose that I is an ideal of the ring k[x_1,\ldots,x_n] and that f\in I. By definition, the locus \mathcal{Z}(I)\subset\mathbb{A}^n is the collection of all points (a_1,\ldots,a_n)\in\mathbb{A}^n for which f(a_1,\ldots,a_n)=0 for all f\in \mathcal{I}. Then, certainly, for all a=(a_1,\ldots,a_n)\in \mathcal{Z}(I), f(a)=0, i.e., f is an element in the ideal \mathcal{I}(\mathcal{Z}(I)) that vanishes on \mathcal{Z}(I). Hence, f\in I implies that I\subseteq \mathcal{I}(\mathcal{Z}(I)).

(10) First, suppose that I is an ideal and that V=\mathcal{Z}(I) is an affine algebraic set. It suffices to show that V=\mathcal{Z}(\mathcal{I}(V)) by way of two-sided inclusion. To that end, let a=(a_1,\ldots,a_n)\in V. Then a is in the zero-set of the ideal I, whereby it follows that f(a)=0 for all f\in I. But for any f satisfying f(a)=0 for arbitrary a\in V, f\in\mathcal{I}(V) and a\in\mathcal{Z}(\mathcal{I}(V)) by definition. Hence, V\subset\mathcal{Z}(\mathcal{I}(V)).

Conversely, if a\in\mathcal{Z}(\mathcal{I}(V)), then f(a)=0 for all f\in\mathcal{I}(V). But all such functions f disappear for all values v\in V=\mathcal{Z}(I) by definition of \mathcal{I}(V). This means that the polynomials f\in\mathcal{I}(V) for which f(a)=0 are precisely the functions which satisfy f(v)=0 for all v\in{Z}(I). Hence, a itself must be an element of \mathcal{Z}(I)=V, whereby the equality is proved.

The other expression is proved similarly and is omitted for brevity. Therefore, as claimed,

\mathcal{Z}(\mathcal{I}(\mathcal{Z}(I)))=\mathcal{Z}(I) and \mathcal{I}(\mathcal{Z}(\mathcal{I}(A)))=\mathcal{I}(A),

from which it follows that the maps \mathcal{Z} and \mathcal{I} are inverses of one another under the construction given here.    \square

Dummit and Foote Example

I found one particular example – namely Example 2 on page 660 of Dummit and Foote – to be a good exercise in the sense that it tied together a lot of ideas from earlier parts of the book. In order to share, though, I need to give a little background.

Throughout, let k denote a field. The affine n-space \mathbb{A}^n over k is the set of all n-tuples (k_1,k_2,\ldots,k_n) where k_i\in k for all i. For a general subset A\subset \mathbb{A}^n, the ideal \mathcal{I}(A) is called the ideal of functions vanishing at A and is defined to be

\mathcal{I}(A)=\{f\in k[x_1,\ldots,x_n]\,:\,f(a_1,\ldots,a_n)=0\text{ for all }(a_1,\ldots,a_n)\in A\}.

It’s easily verified that \mathcal{I}(A) is indeed an ideal and that it is, by definition, the unique largest ideal of functions which are identically zero on all of A. With that in mind, consider the aforementioned example:

Page 660, Example 2. Over any field k, the ideal of functions vanishing at (a_1,\ldots,a_n)\in\mathbb{A}^n is a maximal ideal since it is the kernel of a surjective (ring) homomorphism from k[x_1,\ldots,x_n] to the field k given by evaluation at (a_1,\ldots,a_n). It follows that


Proof. As mentioned above, \mathcal{I}((a_1,\ldots,a_n)) is certainly an ideal. To show that it’s a maximal ideal in k[x_1,\ldots,x_n], first define \varphi:k[x_1,\ldots,x_n]\to k by the action that sends f=f(x_1,x_2,\ldots,x_n) to the constant f(a_1,a_2,\ldots,a_n)\in k. Verifying that this is a ring homomorphism is trivial, and given an element x, x=f(x,0,\ldots,0) where f(x_1,\ldots,x_n)=x_1 is a polynomial in k[x_1,\ldots,x_n]. Moreover, the kernel of \varphi consists precisely of those elements in f\in k[x_1,\ldots,x_n] for which f(a_1,\ldots,a_n)=0, whereby it follows that \ker(\varphi)=\mathcal{I}(A) where A=\{(a_1,\ldots,a_n)\}\subset\mathbb{A}^n.

Hence, \varphi is a surjective ring homomorphism with kernel \mathcal{I}(A). In particular, by the first (ring) isomorphism theorem, k[x_1,\ldots,x_n]/\mathcal{I}(A)\cong\text{Im}(\varphi) where \text{Im}(\varphi)=k by surjectivity of \varphi. Clearly, then, k[x_1,\ldots,x_n]/\mathcal{I}(A) is a field, something that happens if and only if the ideal \mathcal{I}(A) is maximal. Therefore, the first claim is proved.

For the second claim, note that the ideal (x_1-a_1,\ldots,x_n-a_n) is an ideal that vanishes precisely on A. Clearly, then, \mathcal{I}(A)\supseteq (x_1-a_1,\ldots,x_n-a_n). On the other hand, an element f\in\mathcal{I}(A) is an element that vanishes at the point (a_1,\ldots,a_n), whereby it follows that

\displaystyle f(x_1,\ldots,x_n)=g(x_1,\ldots,x_n)\prod_{i=1}^n(x_i-a_i)^{\alpha_i}

for some positive powers \alpha_i. Then, clearly, f\in (x_1-a_1,\ldots,x_n-a_n) and so \mathcal{I}(A)\subseteq (x_1-a_1,\ldots,x_n-a_n). This concludes the proof.   \square

Abstract Algebra Quirks

One of the things that has always driven my interest in Abstract Algebra as a field is the perceived multitude of quirks. It’s hard to speak of what I mean from a meta perspective, so I’ll just give an example.

Let k be a field. Recall that a ring R is called a k-algebra if k\subset Z(R) and if 1_k=1_R, where Z(R) is the center of R, that is, the elements x\in R for which xy=yx for all y\in R. From an algebra perspective, any ring R which is a k-algebra is, necessarily, both a ring and a vector space over k. For this reason, when speaking of R being generated by a subset of elements of R, it’s necessary to indicate with which regard the subset generates R.

One cool example of this necessity – an example which I find quirky, in some regards – comes in the form of the polynomial ring R=k[x] in one variable over k. Certainly with this construction, R is a k-algebra, and so R is both a ring and a vector space over k. Note, then, that as a ring, R is a a finite-dimensional k-algebra since x is a ring generator for R – that is, R=k[x] is the smallest ring over k containing x. On the other hand, k[x] has basis 1,x,x^2,\ldots as a vector space over k and hence is infinite dimensional as a k-vector space.

Long story short: Two different structural existences for a single object, and the two are, to some extent, polar opposites as one another.

Things like this always make me realize why algebra is such a necessary and beautiful field.

Algebraic Geometry Realization III: Digging Deeply

Observation III. When trying to learn new things, digging deeply isn’t always the necessary first course of action; sometimes, the information you’re searching out is surprisingly close to home.

This comes on the heels of me realizing something that makes me both irritated and very happy at the same time.

When I was reading through Elliott’s manuscript, I came across a notation with which I was unfamliar. I tried searching through a couple of the sources he cited and found the same notation used several times without ever being fully explained. That made me uncomfortable, because the things being discussed are abstract to the point that much of the intuition stems from being able to decipher what the objects of our structures are, and what the operations acting on these structures actually do.

So I got to digging.

I felt as if the notation (in this case, juxtaposition of structures for which juxtaposition doesn’t immediately make sense to me) was pretty similar to something discussed in elementary ring theory, so I pulled up my digital copy of Dummit and Foote and decided to do some digging.

While digging, I realized something that I’d never realized before: The last part(s) of Dummit and Foote discuss lots of topics I’d never read about, one of which is algebraic geometry! You see, despite my having used Dummit and Foote for a total of four semesters, I’ve never done so in a class that made it to parts 5 and 6 of the text. As such, for all intents and purposes, those sections didn’t even exist in my mind.

This is irritating because it means I missed out on a pretty easily-accessible source of information that I had close by for the past three years, but makes me very happy because I have a source close by that’s pretty easily-accessible! This is definitely a winning scenario for me.

So the lesson for today, kids, is that you should never ever forget what you have close by.