John Ashbery

I own five books of poetry. All five of which are John Ashbery’s (Some Trees, The Tennis Court Oath, Rivers and Mountains, Double Dream of Spring, and Three Poems). I first learned about John Ashbery over six years ago when I wrote a long paper analyzing The Ecclesiast. I wish I knew where that was, since nowadays I can’t make heads or tails of it, and a 15 page analysis would be useful.

Anyone reasonably familiar with contemporary American poetry has probably at least heard the name, since he won the Pulitzer, the National Book Award, and the National Book Critics Circle Award (sometimes known as the “triple crown of poetry”) for his most famous book Self-Portrait in a Convex Mirror. He is one of the first poets in what is known as the “New York School of Poets” which also consisted of Frank O’Hara, Kenneth Koch, and James Schuyler.

The New York School of Poets is the basis of the book I just finished reading The Last Avant-Garde, though I don’t want to bring up the “avant-garde” discussion again. I must admit that I probably haven’t revisited these works in the past six years, which is sort of a shame, but also gives me a fresh clean way to look at them.

The most striking thing that I want to talk about is that back when I first got interested in Ashbery, it was because his poems were “cool.” They weren’t like any poetry I’d ever seen. Of course, what I didn’t realize was that that was the point. So what I learned from the book were a lot of the methods that the NY school used. They had some Dadaist-type influences. They liked the idea of randomness and chance playing an integral role in their poems. In general, the hated the Beats. The Beats were about rebellion and were intense. The NY school were more formalistic.

I think these ideas might be important as a movement, but Ashbery seems to differ. I didn’t notice this six years ago. Ashbery may have claimed to be using randomness, but his poems are far from it. They are also far from this notion that the NY school weren’t concerned with content. His poetry is actually astonishingly content packed. It isn’t even very obscured. I’m not stretching for some deep meaning. It really sort of slaps you in the face:

I thought that if I could put it all down, that would be one way. And next the thought came to me that to leave all out would be another, and truer, way.
clean-washed sea
The flowers were.
These are examples of leaving out. But, forget as we will, something soon comes to stand in their place. Not the truth, perhaps, but–yourself. It is you who made this, therefore you are true. But the truth has passed on
to divide all.
Have I awakened? Or is this sleep again? Another form of sleep?

I can’t seem to get it to format properly. There is more whitespace in there. This is the opening to the huge prose poem The New Spirit. I know I’ve grown in my ability to interpret poetry, but really I think he is being quite clear in this opening. He is laying out (with example) how he is going to write the poem. The meaning is in what is left out. He starts right in after that on the nature of truth and in some sense the nature of consciousness (is life just a dream? how can we tell? things of this sort).

But that may seem sort of cliche or frivolous. Remember that is just the opening to a 31 page poem. And I can literally open up to any part of those 31 pages and get something even more profound (at random):

Nevertheless the winter wears on and death follows death. I’ve tried it, and know how the narrowing-down felling conflicts with the feeling of life’s coming to a point, not a climax but a point. At that point one must, yes, be selective, but not selective in one’s choices if you see what I mean. Not choose this or that because it pleases, merely to assume the idea of choosing, so that some things can be left behind.

This idea that life comes to a point as we get older is certainly not new. The first thing that comes to mind for me is the structure of The Death of Ivan Ilych by Tolstoy. I’ve never heard it phrased that way, though. The entire poem is a carefully phrased meditation on the nature of life and death. That is hardly void of content.

So my thesis of this blog post is just that I think Ashbery should be treated as a very special case of the NY school. Sure his poetry uses lots of neat tricks like self-reference and randomness, but when I hear NY school, I think it is too easy to think that the poet falls far from center on the form vs content scale. One could spend their whole life interpreting the content of this single Ashbery poem.


A closer look at Spec

Let’s think about what is going on in a different way. So now let’s think of f \in R elements of the ring as functions with domain Spec(R). We define the value of the function at a point in our space f(P) to be the residue class in R/P. This looks weird at first, since the image space depends on the point that you are evaluating the function.

Before worrying about that too much, let’s see if we can get this notion to match up with what we did yesterday. We have the nice property that f(P)=0 if and only if f \in P. (Remember that even though we think of f as a function, it is really an element of the ring).

Define for any subset of the ring S the zero set: Z(S)=\{P\in Spec(R): f(P)=0, \forall f \in S\}. Now from what I just noted in the previous paragraph, we get that these are just precisely the elements of Spec(R) that contain S, i.e. the closed sets of the Zariski topology. Thus we can define our basis for the Zariski topology to be the collection of D(f)=Spec(R)\setminus Z(f).

We also will want what is “an inverse” to the zero set. We want the ideal that vanishes on a subset of Spec. So given Y\subset Spec(R), define I(Y)=\{f \in R : f(P)=0, \forall P\in Y\}. Now this isn’t really an inverse, but we get close in the following sense:

If J\subset R is an ideal, then \displaystyle I(Z(J))=\sqrt{J}. Taking the ideal of the zero set is the radical of the ideal. And the radical has two equivalent definitions: \displaystyle \sqrt{J}=\cap_{P\in Spec(R), P\supset J} P=\{a\in R : \exists n\in \mathbb{N},  a^n\in J\}.

If we take the ideal and zero set in the other order we get that Z(I(Y))=\overline{Y} : the closure in the Zariski topology.

We can abstract one step further and put a sheaf on D(f). Note that for any f\in R we have that \{1, f, f^2, \ldots\} is a multiplicative set, so we can localize at it. Since I haven’t talked at all about sheaves, I’m not sure if I want to go any further with this, so maybe I’ll do some more examples next time and possibly start to scratch this surface.

Spec? You mean like glasses?

So I’ve built up localization starting there, and I’ve built up the theory of prime ideals scattered throughout, but ending here. I also just assume the basics of topology in my posts, so we are in the perfect position to talk about a very fascinating construction and incredibly useful tool that combines all these things.

Warning: I have just started learning about this stuff, so it could be riddled with confusion or error. Luckily, I’m just posting the basics which some readers probably know like the back of their hand and will hopefully point out problems.

Of course what I’m referring to is Spec. As usual let’s assume that R is a commutative ring with 1 (I don’t think we need the 1). Then Spec(R)=\{P : P \  prime \ ideal \ of \ R\}. So we have the collection of all (proper) prime ideals of the ring. Other than prime ideals being my favorite type of ideal, this seems to be useless right now.

Let’s put a topology on our set now (the “points” of our space are prime ideals). Let asubset R be any ideal. Define V(a)=\{\mathfrak{p}\in Spec(R) : a \ subset \ \mathfrak{p}\}. Then we define the closed sets of the topology to be the family of all such sets, i.e. \{V(a) : a \ subset \ R \ an \ ideal\} are the closed sets. This is known as the Zariski topology.

To check that these really satisfy the right axioms, (I won’t go through it, but) note that V(0)=Spec(R), V(R)=\emptyset, V(\sum a_i)=\cap V(a_i) and V(a \cap b)=V(a)\cup V(b) (The last is probably the least trivial, but they all follow in a straightforward from definition way).


1. If our ring is a field k, then Spec(k)=\{*\} the spectrum is a point.

2.Another common example would be Spec(\mathbb{Z})=\{(0), (2), (3), (5), ldots \}. In other words, the prime ideals can just be identified with the prime number that generates them (and we have (0) as a special circumstance). So open sets are subsets of \mathbb{Z} that are missing finitely many prime numbers. So we see that the Zariski topology is not Hausdorff (and rarely is). It will, however, always be compact.

3. Possibly the most important examples are the ones dealing with polynomial rings. In the nicest case, when k is an algebraically closed field, we have that Spec(k[x])=\{*\}\cup k since the prime ideals are just multiples of linear polynomials, we have the bijection of sending any c \in k to the prime ideal generated by (x-c) (and we still have that pesky “zero” floating around that we’ll talk about later).

Last for today is that Spec is a contravariant functor from rings to topological spaces. We’ve basically done everything we need, since we see how it takes a ring object to a Top object. Also if we have a ring hom f:R \to S, then define Spec(f)=f^* : Spec(S)\to Spec(R) in the obvious way, i.e. \mathfrak{p} \mapsto f^{-1}(\mathfrak{p}).

I promised some localization and we should be able to get to that next time, but there is just so much going on here that it is nearly impossible to exhaust (well, from my perspective as a newbie to the topic).

On “Being Political”

I really do want to get back to some actual math that I’ve found interesting recently, but there is one last thing that has been turning in my head recently. It is on the ethics of a certain attitude that has developed in our culture. It is the apathetic attitude toward things labeled as “political.”

My family is a largely non-political family. In fact, they are so anti-political that certain members aren’t familiar enough with politics that they probably would have to take a 50/50 guess at what party Obama belongs to (or Bush, to emphasize that it isn’t a confusion of being close to center).

Don’t get me wrong. I hate “politics” as much as the next person. I hate the lies for political gain. I hate talking around issues instead of about them. Heck. I hate the idea of endlessly talking to stall doing things. I hate that you have to satisfy a constituency instead of thinking what the best option would be. I hate that religion has incredible amounts of influence on decisions that should be rational. And so on….

So here is my claim. It doesn’t matter if you hate or love politics, in any case it is unethical to use this as an excuse. What do you think about the fact that gay marriages could be repealed? Oh, that’s just a political thing. I don’t really follow that. What about the genocides in Darfur? Well, there isn’t really anything we can do about that. Oh, you went to that protest? That was pointless. It is political and the politicians aren’t affected by protests. The examples are endless.

This idea that because something can be linked to politics (or that it is something that politicians will have to vote on) is hopeless to change and hence a waste of time and money to try to change is unethical. It has become an excuse to stand by and watch inhuman things happen without feeling guilty. The main problem is that people that do this are completely unaware that this is what they are doing. It isn’t really their fault. This idea has been culturally accepted, and culturally reinforced when we see all the scandals and lying going on.

I guess my main point is that this cultural acceptance needs to start to turn around if we really want to see positive changes. The next time you hear someone turn something down or refuse to comment or act on something for this reason point it out. The best way is to head-on point out that there is a moral issue that is not political that they are standing by and letting happen and just using that as an excuse. I mean, what can’t be considered “political”? The statement, “That’s all politics,” is really void of meaning when you think of it that way.

An Avant-Garde Movement for Math

This idea has been tossing around in my head recently, since I’ve been reading the book The Last Avant-Garde: The Making of the New York School of Poets. I know I’ve done the art/math comparison before. I think I’ve even done the math education/art education comparison before. But I want to re-cap it and go one step further this time.

I don’t really want to make the arguments that math is an art form. It is a rather easy argument to make (the hard parts are creating an aesthetic theory, etc). The comparison this time is that of education. I find it extremely disconcerting that math and science are taught so differently from art. Art education tries to bring out the original ideas in the student so that new and exciting progress can be made, whereas math and science education tries to squash creativity and teach that there are only right and wrong answers (at least in the American primary/secondary/undergraduate system).

Now I hear the complaints already, “But there are right and wrong answers in math. There is no such thing as right and wrong in art. Anything goes.” And so on. But hold on a minute. The person that makes these claims has certainly never taken music lessons. There are just as many black and white correct and incorrect ways of doing things. The rhythm is either right or wrong. You’re either in tune or you aren’t. You are either playing the right note or you aren’t. You must rigorously train all of these mechanical techniques just as mathematicians must train the mechanical techniques of symbol manipulation and logical argument.

The problem is that in art education, the student is constantly reminded that these mechanics must be learned, but they are not the art itself. Once you’ve internalized the mechanics you must transcend them in order to create new original art, but it is only after the necessary evil of these basics that one can do this.

In math education, we face the problem that not only is the second step of the art not emphasized, but it is often never even mentioned. You must learn basic arithmetic and how to solve quadratic equations, but this is not math by any mathematician’s standards. In order to create new and original math, you have to transcend these basics just as the artist has to. This is much harder for mathematicians (in my opinion) solely due to this educational reason. We get indoctrinated with old methods and techniques, and never are explicitly reminded that new techniques need to be invented. It is the creation aspect of math.

I don’t want to dwell on that since I believe I’ve posted about it before (I could go on forever about it), but I did want to bring up those points again so that the main idea here has context. All other arts have had “avant-garde movements” in some sense. These avant-garde movements have essentially rendered it nearly impossible for truly original things to come out of the arts now. The rules have been followed, the rules have been completely broken, and now if you sit somewhere in between it can be said to be a conglomeration of the two. The importance of avant-garde movements should not be underestimated, though. The avant-garde has opened up the freedom to create precisely what you want, and it will have a context. Creativity for these arts is also now a two-way street and not just a one-way as before the movement. Creativity comes not only from pushing the limits of the rules, but also from toning down the lack of rules of the avant-garde.

This is what I propose. There has been no avant-garde mathematical movement (well, maybe…). One reason that it is hard to produce original math is that mathematicians constantly have to push out in creativity. An avant-garde movement always will open up the door for originality by toning down. It cracks open an avenue for incredibly new techniques that can produce lots of results.

So why did I write “well, maybe…”? I think there have been some potential quasi-avant-garde movements already, and they have been wonderfully fruitful as I said should happen. I think the idea of categorification, or category theory as a foundation for mathematics as opposed to set theory was a pretty radical shift in perspective. I think anytime a major shift in notation useage happens, this could boarder on the avant-garde as with string diagrams.

The one true avant-garde that first got me thinking of this, though, is Grothendieck. I truly am not qualified to talk about any of his work (I’ve only recently started learning about schemes even), but it completely revolutionized algebraic geometry, not because of the standard mathematical method of proving/improving theorems. Not because of the standard mathematical method of inventing new techniques and tools. But because he actually completely changed the way people view and think about the subject. In the way of the avant-garde he decided to throw out not just some, but all of the old rules and invent his own. Now this can be applied all over mathematics.

So when I call for an avant-garde movement in math, I don’t mean throw out all of the rules in the sense of logic and sound reasoning, I mean dare to think radically differently than your predecessors. After all, the avant-gardes of music still used sound and instruments, the poets and authors still wrote things using words and English, painters still used paint. Avant-garde doesn’t mean you stop using what foundationally makes your art identifiable, but it might render some branches as unrecognizable (I don’t think a pre-Grothendieck algebraic geometer would recognize current algebraic geometry as such).

Anybody in?

Best and Worst of 2008

It’s that time again. This will probably be a long post, but I’ve spent a lot of time over the past week or so preparing this (estimating about an hour per album, then 37 hours of listening).

Top 10 albums of the year:
1. For Emma, Forever Ago by Bon Iver
2. Some Are Lakes by Land of Talk
3. The Slip by Nine Inch Nails
4. Punch by Punch Brothers
5. At War With Walls and Mazes by Son Lux
6. Third by Portishead
7. Visiter by The Dodos
8. Falling Off the Lavender Bridge by Lightspeed Champion
9. Oracular Spectacular by MGMT
10. Secular Works by Extra Life

Now to make this easier, basically every major top albums of the year list has already come out, so I know what is strange and what isn’t. Bon Iver, Portishead, the Dodos, and MGMT get at least nods from everyone. Land of Talk isn’t too surprising considering Justin Vernon produced it. NIN probably needs some justification. Well, I’m not a NIN fan, but this album released for download for free, so I said, “Why not?” Imagine my surprise when it blew me away. Lyrically it is quite introspective and philosophical. The songs range from hard rock, to electronica, to beautiful soft soundscape. It is truly an album by a virtuoso.

On the other end of the spectrum are the Punch Brothers. They are sort of like a bluegrass band, yet fully compose their music. The first four tracks are a “suite,” and are more intricate and fully developed than most classical music I know (and I started college as a music major). It really deserves the number 4 spot, or maybe even higher.

I’ve raved about Son Lux before. His music is sort of like NIN, but more approachable. The only reason he is lower is that lyrical basically nothing happens. Lightspeed Champion is really amazing and did get some recognition on lesser known lists. He basically is what would happen if you had someone trained in rap and death metal have a revelation and start writing country music. Yes. It is the most unique sounding band on the list. Lastly, Extra Life has the most sophisticated artistic vision on the list. This group uses Renaissance and earlier melodies and incorporates it into modern “math rock.” This one will probably take the most listens to start to understand.

Honorable mentions:
Blue Lambency Downward by Kayo Dot
Fleet Foxes by Fleet Foxes
Dear Science, by TV on the Radio
Keep Your Eyes Ahead by The Helio Sequence

I don’t think anything needs justification there.

Bottom Five:
Conor Oberst by Conor Oberst
The Airborne Toxic Event by The Airborne Toxic Event
808s and Heartbreak by Kanye West
Perfect Symmetry by Keane
Liferz by Blood on the Wall

Let’s start with Blood on the Wall. I only got this because half way through the year, people who had nearly exactly the same list as me also had this. Big mistake. The thing I dreaded most about listening to the year again was the fact that I knew I had to listen to this garbage again. I almost deleted it from my ipod to pretend like I never got it in order to not have to review it again. Keane usually does stuff I like, but this time the lyrics are horrifically cliche and the music your standard pop. Even worse, they try to spice it up with all these random electronic effects which just makes a bad pop song near unlistenable.

The Kanye album is boring…and bad. The Auto-tune thing sounds very outdated and is used on every track. The background music is even worse with minimal drum machines often sticking out like a bad remake of cliche ’80’s music. Most surprising of all is Conor Oberst, though. I love him usually. Again with the awful lyrics. I never use lyrics to judge an album as I am much more musically than verbally inclined, but when they are this bad it is unavoidable. “He’s gonna DO IT. He’s gonna DO IT. He’s gonna DO IT by hand.” (referring to a post man delivering a letter). Or the upbeat country song in which the phrase “I don’t wanna die in a hospital” is repeated almost exclusively.

Enough of that.

Albums that were evaluated that fell somewhere in between:
Pretty Odd by Panic at the Disco, Some People Have Real Problems by Sia, The Seldom Seen Kid by Elbow, Modern Guilt by Beck, Quaristice by Autechre, Lost Wisdom by Mount Eerie, At Mount Zoomer by Wolf Parade, The Stand-Ins by Okkervil River, Viva La Vida by Coldplay, Another World EP by Antony and the Johnsons, Water Curses EP by Animal Collective, Volume One by She & Him, Rearrange Us by Mates of State, Narrow Stairs by Death Cab for Cutie, Attack and Release by The Black Keys, Vampire Weekend by Vampire Weekend, Lucky by Nada Surf, You & Me by The Walkmen

The attentive reader will note that it is not in alphabetical order and thus it actually is in the order that I ranked them (Pretty Odd being the highest non-honorable mention and You & Me being basically awful but not bottom 5).

As for the top 10 individual songs:
1. “Midnight Surprise” by Lightspeed Champion
2. “Head Down” by NIN
3. “Raise” by Son Lux
4. “Future Reflections” by MGMT
5. “Give Me Back My Heart Attack” by Land of Talk
6. “The Wolves (Act I and II)” by Bon Iver
7. “The Season” by The Dodos
8. “Broken Afternoon” by The Helio Sequence
9. “DLZ” by TVotR
10. “The Bones of You” by Elbow

Props go to The Helio Sequence, TV on the Radio, and Elbow for not having an overall top 10 album but having some great individual song. Also note the placement of Bon Iver at 6 despite having the number 1 overall album. I found this interesting. You just can’t beat Midnight Surprise. Look it up on youtube or something. It is an epic (something like 8 minute) song spanning all sorts of genres. Constantly changing keys and tempos and time signatures and styles and textures. Yet it all continues to flow. I hadn’t listened to the album as a whole in probably 8 months, so I was shocked that I forgot how great that song was and had to make it number 1.

Note that many great bands were not reviewed due to time and monetary constrictions. If you have a top band that you think I overlooked please comment! (I’ve gather from other lists that Deerhunter and My Morning Jacket I have to check out).

Finite Groups as Galois Groups

So my old proof isn’t really working on wordpress for some reason, so I’ve taken it as a sign to do it in a different way. This method is far more complicated than the old way (in which I just call upon some theorems and look at orders and then am done), but I think it better gets at what is going on.

Anyways, since we were on the topic of Galois theory, here is a fact I found astonishing the first time I heard it (maybe it is quite obvious to you). Every finite group arises as the Galois group of a field extension, moreover we can choose the two fields to be number fields. Recall that a number field is just a subset of the complex numbers that is algebraic over \mathbb{Q}.

Proof: Let G be a finite group. Then by Cayley’s theorem G\cong H< S_n for some n. But then there is a prime p, such that n<p, meaning S_n<S_p. Let’s find a Galois extension K/\mathbb{Q} such that S_p\cong Gal(K/\mathbb{Q}).

Our natural choice is the splitting field of f(x)=x(x^2+1)(x^2-1)(x^2-4)\cdots (x^2-m^2)+1/p, (note the sarcasm) where we chose our prime to be of the form p=2m+3. Thus we have that deg(f(x))=p, it is irreducible by Eisenstein’s criterion, and it has exactly 2 roots in \mathbb{C}\setminus\mathbb{R}. Thus, Gal(K/\mathbb{Q})\cong S_p. The details of this just amounts to playing around with cycles.

Now we can just invoke the Fundamental Theorem of Galois Theory. We have the subgroup H < S_p which corresponds to the fixed field, say L, where K\supset L\supset \mathbb{Q} and Gal(K/L)\cong H\cong G, which is what we wanted.

Now that this is typed out, I think the other way is better since I didn’t have to skip over the cycle argument. This method is just as mysterious overall, but in fact the other is probably more believable, since I only use things that are standard.

Lastly, let me just clear up some mystery about f(x) at least. The way we know it has 2m+1 real roots is that f(x) alternates signs between pairs -m-1/2, -m+1/2, then -m+3/2, -m+5/2, all the way to m-1/2, m+1/2. Thus by the IVT, there is a real root between each of those. Thus there are 2 complex roots left over. We can even locate them to be near \pm i if p is sufficiently large by Rouche’s Theorem. Also, f(x) is irreducible iff pf(x) is irreducible, and pf(x)=px(x^2+1)(x^2-1)\cdots (x^2-m^2)+1 and here it is clearly seen that p divides all the coefficients except the constant term and p^2 does not divide the highest term (Eisenstein).

Oh, and while I’m at it, I might as well lead you in the right direction if you want to check that Gal(K/\mathbb{Q})\cong S_p. By order, we know that Gal(K/\mathbb{Q})\leq S_p (at least there is an isomorphic copy in there). So let the copy of Gal(K/\mathbb{Q}) act on S_p. The action is transitive, so there is a 2-cycle, and since p divides the order there is a p-cycle. These three things should get you there (that it is inside S_p, that there is a 2-cycle, and that there is a p-cycle).

So although they exist, they aren’t always the easiest to find. In fact, if we want our base field to be \mathbb{Q}, then this is known as the “Inverse Galois Problem” and is still open. Some cases have been resolved. For instance, it is known that every finite solvable group arises as a Galois group of an algebraic extension of \mathbb{Q}.

Weird Bug

This picture thing turned out to be a pain. Maybe there was some easy fix, like, I don’t know, clicking “Use your old picture,” but that didn’t seem to work. So then I just tried re-uploading it, but the size was too big. Then no matter how I shrunk the size and dimensions, when I would re-upload it would just blow it back up to the original size. Thus, I had to change the picture to one in which I was further away from the camera in order to have a head small enough so that after cropping you could see more than just a single eye.

I have this nice post all typed up on every finite group arising as the Galois group of number fields, but there is some weird LaTeX bug that is cutting out some sentences. I also have a square bracket problem, which I think I’ve narrowed down to: you cannot start your LaTeX code with “[“. The other weird thing might be that you cannot end with “!”?

Both of these are bad considering I’m dealing with the group S_n, so it has order n!, and I’m dealing with field extensions which means I need [K : F], but I’ve subscripted, so I need it inside of LaTeX code…

Descent Theory

I’m going to do a change in plan.

Galois Theory: Let F be a field. In some sense the “universal” Galois group is Gal(\overline{F}/F) where \overline{F} is the algebraic closure, since given any algebraic extension K/F we have that Gal(\overline{F}/K) < Gal(\overline{F}/F). In fact, there is a bijective correspondence between subgroups of the Galois group and algebraic extensions (this is just loosely speaking to show a connection later on, I’m not being careful about finiteness and things). In this case the we have an inverse corrolation. As the fields get bigger, the groups get smaller.

Covering spaces: For suggestive notation, let’s denote Y/X to mean Y is a covering of X. Then if X has sufficiently nice conditions, we have that there is a universal cover \overline{X}/X with covering map q: \overline{X}\to X. Then we have that Aut_q(\overline{X})\cong \pi_1(X) where Aut_q(\overline{X}) is the group of “deck transformations,” i.e. the automorphisms \phi: \overline{X}\to\overline{X} such that q\circ \phi= q. Now any other cover will “sit below” the universal one, in that the covering p: Y\to X will have a factoring \overline{X}\to Y\to X. Moreover Aut_p(Y)\cong H<\pi_1(X). Just as in the Galois case, there is a bijective correspondence between conjugacy classes of subgroups of \pi_1(X) and isomorphism classes of coverings. This time in a sense it is not reversed, though it depends on how you want to look at it.

I found the similarities of these two situations very strange. There must be something deeper. All field extensions are in bijective correspondence to subgroups of the Galois group of the “largest one,” and all (iso classes of) coverings are in bijective correspondence with (up to conjugacy) subgroups of the fundamental group.

It turns out that after some hunting, there is a huge deep field called “the theory of descent” or something similar. It all looks so fascinating, but it is just too far astray from what I’m studying for me to actually learn right now. I thought I could dip a toe in or something and report back my findings, but there doesn’t seem to be any good introductions to the subject or any hope for quickly seeing some of the ideas. So, after a few days of hunting, I’m changing my plans and am going to look for something new to go on about (possibly back to the prime and localization that I built up, then left for dead?).

Actually, if anyone knows of a place to learn some of this stuff, it would be greatly appreciated if you let me know!