Differential Geometry -> Algebra -> Clifford Stuff -> Algebraic Topology (optional),

and since yesterday was (unofficially) differential geometry day, I’m going to spend today doing algebra.

First order of business: Eisenbud and Harris. And, since I’ve been meaning to write down some of the solutions to exercises I’ve passed, I guess I’ll do that here.

I-1. Find when is:

1. 
2. 
3. 
4. 
5. 
6. .

Proof. Recall that the elements of the set are the prime ideals of and that the 0 ideal is included (I’ll not write 0 hereon except when absolutely necessary). Therefore, for part 1, is the set of all where is prime. For (2), note that is a field, that every field is a PID, and that in a PID, maximal ideals and prime ideals are equivalent; moreover, note that in , for all integers relatively prime to , and because all integers are relatively prime to the prime number 3, .

On the other hand, isn’t a field – or even an integral domain – and so the possible elements of are of the form for (0 is already mentioned above, and due to 1,5 being relatively prime to 6). Note that isn’t prime since for . Simple computation verifies that are prime in , however, so . Because , part 4 is equivalent to part 2.

For part 5, being a field implies that is a PID, so it suffices to find all maximal ideals there, and because is a field (for a PID) precisely when is -irreducible, it follows that

.

Of course, is algebraically closed, whereby it follows that the collection of such ideal-generating polynomials is precisely the linear ones, namely those of the form , . Finally, note that consists of elements of the form for . Ignoring units, this ring therefore consists precisely of the linear polynomials , , and because of part 5, it follows that .  

I-2. What is the value of the “function” 15 at the point ? At the point ?
Proof. The notation here is as follows. Elements are considered as functions on the space whose valuations are computed as: If , the quotient field of the integral domain consists of values defined to be the image of via the canonical maps . So then, because the quotient field of is itself (since is a field), the value of the “function” 15 in is simply . Similarly, is again a field, whereby the value of 15 there is given by .  

I-3. (a) Consider the ring of polynomials , and let be a polynomial. Show that if is a number, then is a prime of , and there is a natural identification of with such that the value of at the point is the number . (b) Show that this holds in the general case as well.
Proof. As demonstrated above, the fact that is a PID and that is a field for all shows that is prime. Moreover, because the image of in is defined to be the image of in the quotient field of – that is, itself, since is maximal – the identification given is derived equivalently to the derivation of the fact that by way of the correspondence .

In the general case, is the coordinate ring of an affine variety over an algebraically closed field and is the maximal ideal corresponding to a point in the usual sense, whereby the conclusion is that and is the value of at in the usual sense. This result follows from a construction identical to that in case (a) due to the fact that being maximal implies that is a field that’s therefore equal to . Hence, because , corresponds to and corresponds to .  

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.

Among my recent failures in differential geometry are my oft-written-about issues with Kobayashi and Nomizu, which led of course to beginning to look at books by Auslander, Wells, Warner, and Lee (this one and this one). I never really gave Auslander a fair shake before that one ended up on the bottom of the stack; Wells’ book is disappointing because the later material is precisely what I’d be looking for in terms of combining machinery from different subdisciplines, but by trying to generalize the behavior of differential, real-analytic, and complex-analytic manifolds, the notation is clumsy and unintuitive. I like Warner overall but the exercises seem pretty well beyond the scope of what the text addresses, and, by and large, the first of Lee’s books listed there is good but is more topology and less geometry. The second of his books is more geometry, but after dredging through 40-ish pages in a day, I hit new material with the realization that I didn’t know the requisite material very well.

Today I decided once and for all to suck it up and start working through Spivak’s big book(s). I cranked out about 40 pages there today – even making progress on some exercises – before calling it quits. My summer-long suspicion that I’d be behooved to start working through his small book as well was confirmed, so I think those two + Warner will constitute my differential geometry plan moving forward.

Then, there’s algebra. I need to work on the commutative algebra book; I need to learn algebraic geometry. I know I need to focus on Eisenbud/Harris and on related resources needed to acquire that knowledge successfully, but I can’t seem to shake the voice in my head that tells me to focus on Dummit and Foote first. That uncertainty is furthered by the lack of specifics received from faculty members I’ve emailed. Oh well.

I also haven’t been able to hash out a clear avenue of communication with the professor whose Clifford analysis paper is now an increasingly-sparse, overwhelmingly-frustrating and difficult part of my routine. There really is no excuse for me being as slow as I have been on that front, and the more I progress, the more I realize I’m being forever shitted on by my having never taken a graduate-level linear algebra course. I have a rather highly-regarded text downloaded for that purpose; perhaps that should also somehow find its way into my routine.

Guess I need to make zero hours become not zero hours. There’s a thesis topic for me to ponder.

Somehow, too, I’ve not mentioned Hatcher or algebraic topology in general, nor have I begun to fret about my summer Foundations of Mathematics class which begins one week from today.

The more I stare at this screen and keyboard, the more depressing it becomes.

So that’s it…first things first moving forward will be to devise a study plan. That’ll be the beginning of tomorrow for sure.

Until next time, au revoir!

This is how I’m spending my summer vacation: Graduate Texts in Mathematics.

I might have a problem.

I regret to inform: That is not the case.

Much to my surprise, I actually did carry out the plan outlined in my previous entry: I awoke at 8am, ran an errand or two, and made it to my office at about 10am. While there, I was hoping to get some solid reading done and to run into a professor about a resource he may be providing me; I left having only done some semi-solid reading. Despite spending some 3 hours there, I managed to only read a handful of pages of the aforementioned professor’s paper on Cliffordian analysis and -conformal mappings plus the remainder of Dummit and Foote’s treatment of Gröbner bases, though in my defense, “reading” at this point consists of reading a statement, trying to believe that it’s true, and trying to write down the details of said truth with no outside influences.

So like has become my mantra: I didn’t do much, but I did very little nothing.

At this point, my mind is too sleep-deprived to remember if I did work at home afterwards. I know I’ve been nose-deep in so many different pieces of literature on so many different days that all I really have almost no ability to differentiate topics, authors, days, times, or activities. The one thing I do know is that I feel my learning growing without feeling as if I’ve done any work.

Now I’m rambling in circles.

Fast forward to Tuesday and you’ll get even less than you got Monday: I literally did no math whatsoever until almost midnight. Then – which was about 3.5 hours ago, now – I decided to spend some time ensuring that I at least partially fight away the ring-rust, and have since been reading up on Function Algebras (via this paper), Schemes (via Eisenbud and Harris), and Differential Stuff (via Warner). Edit: I remember now that I did do work on Monday evening: I jumped into Wells’ treatment of Differential Analysis on (Complex) Manifolds, which I’ll indubitably talk about before long.

Today’s lack of progression is somewhat disturbing, naturally, but I’m at least partially okay with it due to the fact that I spent some of my pre-midnight mathematizing catching up with old friends/colleagues. Indeed, when I was in Ohio, I was surrounded by people who were, for all intents and purposes, colleagues, but very few of them ever became what I’d consider friends. All said, I made approximately a handful of genuine friendships there, and tonight, I got to reconnect with two who I’ll call L and R.

L and R are, in some ways, at opposite ends of their journeys through mathematics. L is an old-timer, having been around the block more times than R and I combined having invariably forgotten more mathematics than I personally will probably ever know. He’s in the stages of finishing up his dissertation research – and all the struggles that go into that – and so his narration is particularly insightful as far as letting me know what my own future will likely hold. R, though, is at approximately the same level as I: We both started our programs at around the same time, and with only a few minor differences, our careers are pretty much carbon copies of one another. He and I are both wet around the ears when it comes to our careers in academia, and while that itself seems to offer nothing to me in terms of conversation, I find that his perspective and – for lack of a better term – free-spiritedness is something of a motivator. He’s thirsty but not dying of thirst; he’s hungry but not starving; he’s driven, but not driven out of control. He has my good career qualities without succumbing to my pathological career obsession.

So yea…L and R are always good people to reconnect with, and tonight brought me that. Given the fact that I can make up tomorrow for the lack of work today, I’ll say my Tuesday, all in all, wasn’t really so terrible.

Of course, this wouldn’t really be worth blogging about if I didn’t at least bring some math into the picture, so for that, I leave you with this beauty, from Osborne’s treatment of (Basic) Homological Algebra:

Gadzooks!

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 -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….

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.

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.

Per my earlier entry: Given a smooth complex algebraic variety , I finally manage to track down a semi-manageable definition for the structure sheaf . 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 with the claim that …

…say what now?!

Finally, I reread a bunch of stuff and found a cool analogy between arbitrary varieties and the case where is a smooth -times continuously differentiable manifold of dimension . This cleared things up for me.

It really is going to be a long summer. Heh.

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 of complex polynomials in 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 -modules.

le sigh

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 being monogenic: My original claim was that for monogenic functions, but today, I miscalculated the partials for an example that led me to believe that 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 and , respectively, to be a single sum ranging from instead of a term for and , respectively, plus a sum for . Later, when there were two parameters in the sum, I claimed that implies that ; this, of course, is false, since is identified with 1 so that . 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 , an equivalent representation of which is where and where represent the scalar and vector parts of , respectively. In particular, then, if we consider to be the derivative of , it follows that precisely when and . With regards to the scalar part of , this implies that

.

In particular, then, the vector , and so each component must be zero. Hence, for .

If we then turn our attention to the vector part of , we see that , i.e. that

.

Note that the two sums in sum to zero precisely when each sum itself is equal to zero due to the linear independence of the basis elements , . In particular, then, the second sum in 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 cited there should be matrices instead of . Recall also that the first equation in the system of equations shown in the original entry – the equation – is achieved by combining the equation of “mixed partials” of the form from the second sum in with the fact that 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 -module theory. Until next time….

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.

The second thing to keep in mind is that I’m a very ignorant person. I tend to form uneducated judgments under the premises that they’re not at all uneducated, only to find out later that my perception was wrong, that my brain was nothing but ignorant, and that afterwards – invariably – in some way – I’m going to have to suffer as I attempt to either (a) catch up, or (b) give up and mourn.

Now what’s the point of this entry?

Several of my colleagues here are blossoming algebraic geometers: They’re people who recognized the importance of the field (while happening to be good at its constituent components), who identified our institution as one worthwhile in terms of like-minded research fellows, and took the time to uproot their lives to come and be a part of the cutting edge. I happen to be good friends with a couple such people and I can tell you that both are very apt – very very apt, to the point I’m envious most times – and that they’re both very passionate about the research that’s waiting for them in the not-so-distant future.

These two guys aren’t ignorant.

I, on the other hand, came here knowing absolutely zero things about algebraic geometry. In fact, my first exposure to those two words meshed as a phrase was when I was applying to Ph.D. programs: It was then that I discovered that lots of people liked and did and succeeded in research concerning algebraic geometry. Being in my own little world, though, I kept my small-sighted, narrow-minded deduction of areas I knew and liked and wanted to pursue and never bothered learning what all the fuss was about. Then, even when I got here, one of the aforementioned couple took the time to tell me about what he was studying: I knew nothing about it, agreed to participate in an ad hoc reading group he put together, and ended up never being able to attend a single meeting.

So again: I was, am, and continually have been out of the loop.

In this case, though, it’s even deeper than that because in this case, I was truly not interested in the (sub-)area he talked about. One of the things he was most interested in was so-called intersection theory which – as the name suggests – hinges on the study of properties and behavior of intersections of two “curves”. He told me about this and about how his potential advisor was a super-hotshot-rockstar-math-god at this study and as he was talking, I decided that I was really, truly not interested.

And this, folks, is a big deal. You see, in my entire mathematical career (I’m an old man, remember), I’d never once not liked an area of math. I liked some more that others, but I always felt like I’d consider myself a failure for not being able to master all of the areas fully: To me, mathematics is everything and anything less than complete mastery of it means I will have let myself down as a hoarder of mathematical knowledge. Literally, I wanted to know all the math…

…except algebraic geometry…

…and now, here we are.

Since last writing about the analysis of Cliffordian functions, I decided to spend some time researching what I’d been calling Differential Algbera, a phrase I picked up from Ritt’s book of the same name and one that I was (rather incorrectly) throwing around to describe the objects I really wanted to study – namely, rings of differential forms and the behavior of modules on said rings. These are the so-called -modules. After reading some of Ritt’s book, deciding it was irrelevant to what I cared about (sorry, Dr. Ritt), and digging up the two manuscripts I’d downloaded specifically aimed towards -modules, one thing became abundantly clear:

The amount of algebraic background I need to make up before understanding even something basic is probably about the same as the amount of algebraic knowledge I currently possess.

That’s disheartening.

Disheartening, too, are the objects of which I lack knowledge – the objects whose names come up nearly every sentence in any of the manuscripts I found:

Schemes. Stacks. Sheaves of differential forms. Affine varieties.

As sure as I’m writing this entry right now, I can attest that the thing I’m lacking most soundly with regards to being able to study -modules is precisely the field I’d shrugged off ignorantly fewer than two semesters ago….

Ladies and gentlemen: I need to become an (amateur) algebraic geometer.

dims the lights

Admittedly, I panicked a little. Having seen the depth that this field possesses and the amount of effort functional understanding of it requires made me realize that the time to wait ended about five years ago: I have to make progress and fast. One way I started on this path was to accumulate resources: My “Differential Algebra” folder that once housed three lonely, oft-overlooked documents now contains 16 documents of various sizes and depth. I’ve downloaded a few undergraduate-level introductions, and a couple of documents at each of the logical stages in between.

It’s obvious now what I have to do, and so do it I shall…

…starting tomorrow, probably, when I’m not as exhausted, mentally, as I seem to be right now….

So if you guys don’t see me around for the next ten or twelve years, you know I’m a recluse, living dirty and homeless outside the bookstore of some random top-10 research University and mumbling babble about the category of sheaves over a particular topological space along with their morphisms. You guys can thank Dr. Izzo and one of his countless wonderful, fascinating stories for this image.

I guess that just about does it. Until next time,….

The aim is to post about once a day, with the style being something between popular science and academic coursebook. I’ll try to tag posts accordingly, so it’s easy to tell what audience I’m pitching to. The first few days may be an extremely brief recap of some very foundational material to provide some explanation and background for non-mathematicians.

Occasionally I might discuss/opine/rant about other things, including music, sport, and just why we are getting quite so much rain. I hope this will provide a (necessary?) break from the maths. I’ll happily take requests for a post on a particular topic, but I can’t promise to become an instant expert.

Finally I can’t guarantee that everything I write will be entirely correct on first posting. Some of this material I’m learning for the first time myself, and it might take a couple of iterations before I fully grasp the concepts. If you think I’ve been unclear or don’t understand something, please do comment.

If you are still with me, well done! No more administrative faff, I promise! Have a couple of contrasting YouTube videos for your efforts, here and here.

Mikael Vejdemo-Johansson, from the University of St Andrews, sent in this photo some time ago, saying “we were discussing computational approaches to the topology of algebraic varieties, and sampling issues we were having. A very pleasantly geometric topic that generates a lot of sketches and pictures.”

Let C be a category with objects ob(C), let X be a topological space, and define Top(X) to be the category whose objects are the open subsets of (X) and whose arrows are the inclusion maps between open subsets of (X). A sheaf on X of objects of C consists of

i) for every open subset U of X, an object F(U) of C, and

ii) for every inclusion of an open subset V of X in an open subset U of X, an arrow p_UV:F(U)->F(V), such that

condition 1) p_UU is the identity arrow of F(U),

condition 2) p_UW=p_VWp_UV where p_VW and p_UV are defined,

condition 3) for any open covering {V_i} of any open subset U of X, and for any element s of F(U) such that p_UVi(s)=0 for all i, s=0 and

condition 4) for any open covering {V_i} of any open subset U of X, and for any set {s_i} of elements of F(U) such that p_UViintV_j(s_i)=p_UViintV_j(sj) and s_i is in F(V_i) for all i, there exists an element s of F(U) such that p_UVi(s)=s_i for each i.

Presheaves

The definition of a presheaf is weaker than that of a sheaf.  A preasheaf on X consists of

i) for every open subset U of X, an object F(U) of C, and

ii) for every inclusion of an open subset V of X in an open subset U of X, an arrow p_UV:F(U)->F(V), such that

condition 1) p_UU is the identity arrow of F(U) and

condition 2) p_UW=p_VWp_UV where p_VW and p_UV are defined.

Let k be an algebraically closed field.  Affine n-space over k is the set of all sets (a₁,…,a_n) of n elements of k, and is denoted by Aⁿ_k or just Aⁿ. Each element P=(a₁…,a_n of Aⁿ is a called a point, and the a_i are called its coordinates.

Suppose f is a member of the polynomial ring A in n variables over k and P=(a₁,…,a_n) is a point in Aⁿ. Then f(P) is defined to be equal to P(a₁,…,a_n).

Let T be some subset of A. The set Z(T) of zeros of T is defined to be the greatest subset of elements P of Aⁿ such that f(P)=0 for all f in T. The sets Z(T) are called algebraic sets, and the Zariski topology on Aⁿ is defined by taking the closed sets to be the algebraic sets.

A subset of a topological space is irreducible if it is nonempty and contains no two proper closed subsets of which it can be written as the union.

Projective n-space Pⁿ over an algebraically closed field is defined as the set of equivalence classes of elements of Aⁿ⁺¹-{0,…,0} defined by the following equivalence relation: P₁ is in the same equivalence class as P₂ iff P₁=LP₂ for some L in k-{0}. For any member of the polynomial ring in n+1 variables over k and any member e of Pⁿ, define f(e) thusly: select some point P in Aⁿ⁺¹ whose equivalence class is e. Then compute f(e). This value will be determined up to multiplication by a nonzero element of k, which means that nonzero values of f(e) are undefined. Convention remedies this unfortunate situation by defining the nonzero values to be equal to 1.

Varieties
Varieties fall into four types, as detailed below.

Let Aⁿ have the Zariski topology. The closed irreducible subsets of Aⁿ are known as affine algebraic varieties or just algebraic varieties, and quasi-affine algebraic variety or quasi-affine variety is an open subset of an affine variety.

Let T be some subset of the polynomial ring in n+1 variables over k. Z(T) is then defined to be be the set of all equivalence classes in Pⁿ such that f(e)=0 for all f in T, and the Zariski topology can be defined in the same way as for affine space. Let us endow Pⁿ with the Zariski topology, and define a projective algebraic variety or projective variety (respectively, quasi-projective algebraic variety or quasi-projective variety) to be an irreducible closed (respectively, an open subset of an irreducible closed) subset of Pⁿ.

[Note: there also exists the (obviously) less concrete notion of an abstract variety.]