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Long story short: childhood spent writing code -> descent into online gaming as a result of uninspiring course material -> poor marks interpreted by teachers as inability to keep up -> sudden realization on my part that this is happening -> last-minute frantic display of sentience -> aspirations of working on quantum computers in late high school -> somewhat bleak admission prospects to nanotechnology engineering program -> miraculous admission to aforementioned program -> ironic decision to enter dramatically less competitive physics program instead -> exposure to real math -> transfer to pmath/CS -> transfer to pmath/CO. That last arrow, or at least the change it brought about, is important. There are more links in this chain, but it is already long enough.

Minimalism adequately describes the approach I have taken towards many aspects of life: living arrangements, social behaviour, food, clothing, etc. When I moved into a new house (or rather, “mansion” as it came to be known; best house ever, by the way) just over a year ago, the contents of my room (besides essentials like clothing) consisted solely of a bunch of books, a mattress, and my laptop. It remained this way until a friend offered me a desk and bed frame. I accepted because neither effort nor payment was required on my part. I do not value material possessions, and lest I be labelled a reprehensible hypocrite as I sit here typing this post on my MacBook Pro, let me suffix that with “at least relative to many others in similar circumstances”. Fast Internet access is a game changer for me; it is probably the very last material possession I would give up (after all, it certainly subsumes physical books *HOISTS ENORMOUS RUSSIAN FLAG*, although I still prefer reading the latter to staring at a computer screen; call me old-fashioned).

To mention another example, I rarely wish people happy birthday, Christmas, Hanukkah, Ramadan, etc. nor am I offended in the slightest when my negligence to do so is reciprocated by others. There are a few reasons for this; in general these events bear little significance to me. Without even mentioning the arbitrariness of the units of measurement set forth by the Western calendar, many things people celebrate are specific to a certain religion, nation, culture, and so on. Although I do not denounce any of these provided they remain within reason (extremist sects that advise senseless mass murder and so on certainly fail this criterion), I do not particularly favour any of them either. I am Canadian because I was born here, not because I think Canada is the best country in the world. In fact, expressions of nationalism tend to irritate me, since I perceive them as egotistical. I speak English merely because of my initial conditions, not because I consider it better than other languages; conveniently, however, it seems to currently be the language of academia. However, I am not Christian; that is where the influence of initial conditions begins to wane. In fact, I am not religious at all, since no such belief system seems to me any more plausible than the others, and they tend to be pairwise incompatible. If I chose to be religious, why would I choose to be a Christian rather than, say, a Muslim or a Shintoist? Such questions seem to admit no satisfactory answer, so I rest my case.

In any case, I said I would talk about what math has given me, so let me do that now. My decision to study pure math was made with infinite confidence and zero doubt, and really, I have never once regretted it. This can be said about very few decisions I have made. The clarity with which it has made me able to reason and formulate thoughts is indispensable, and I can hardly understand how anyone (much less a scientist) gets away without it.

To my mild surprise, I recently noticed my developing ability to read and understand technical material at a faster pace, for longer periods of time. Teaching myself things seems to have become dramatically easier over the past few years; perhaps this evolved out of sheer necessity, considering my horrendous sleeping habits and often poor lecture attendance, even in graduate courses. As a result, whenever I consume any kind of media lately that is not a Springer GTM, I find myself floored by the comparatively low information density, as though someone suddenly decided to cut the frame rate of life in half.

Perhaps as another consequence, I have been developing interests in other areas, often choosing to spend time learning about more applied areas (statistics, computer science, engineering, physics, chemistry, even history) rather than bury myself in increasingly abstruse pure mathematics (cosheaves of Hilbert C*-modules, idempotent semisymmetric quasigroups, or what have you). This brings me back to my decision a year or two ago to drop my CS major. In itself it is fairly irrelevant, amounting only to the loss of some formal designation; what really *is* significant is that I actually stopped studying computer science (dropping a major and discarding an interest are two very different things). This was likely a mistake, which I justified by claiming things like “there are too many arbitrary details to memorize in anything other than pure math” and naively downplaying the importance of different kinds of knowledge. However, the Platonic realm was a kind of adrenaline, the likes of which I had never experienced, and in retrospect it simply “drowned out” everything else. Now I have sobered up, in a sense, and it seems that at the time I misinterpreted this as a genuine disinterest in other disciplines.

A part of me feels sad to have “wasted” these years, enrolling an unreasonable courseload every term (“Hey, look at all the cool courses offered next term! I will get killed if I enrol in all of these, but I’m going to do it anyway because I don’t want to miss out” -> essentially no free time all term, amidst constant onslaught of assignments etc.) pursuing very few interests apart from abstruse math, but another part of me says it was necessary to get to where I am now, and I should just begin this new phase with no regrets.

I do however have to admit that I consider myself a notoriously difficult person to entertain. Based on my observation, many people are content with lives that I would find insufferable: watching television, listening intently to the constant white noise of Hollywood and the lives of celebrities, social media, distracting themselves with chores, and so on. I thought that mathematical research might be my answer to the “big question”. So far, academia has not turned out to be the atmosphere I imagined (though I suppose this largely depends on location; if your friend circle, say, consists of a bunch of extremely competent mathematicians, and you all hang out together all the time and talk math, then yeah, of course you guys are going to end up doing great things — compare the similarly collaborative nature of many successful Silicon Valley startups). To be specific, it is a little dry, in the sense that it is too much sitting at your desk for hours on end, staring at some cryptic paper, and not enough learning from others. What might take me a whole afternoon to gleam from a paper can probably be *very* adequately explained in under an hour by a knowledgeable colleague. Perhaps this is just an artefact of my clumsiness at meeting people, but I have to say I sincerely doubt it. After thinking about prospects after graduation, I know for sure that I need to move to a big city, since that seems like my best shot at finding the massively collaborative intellectual environment I am after — *precisely* the “hacker culture” from which I may ironically have inadvertently ejected myself by giving up on programming a couple of years ago.

It is morning now, and I think that was essentially all I wanted to say anyway, although I could probably elaborate more on a few points.

Questions and comments welcome, and uh… ahem…

“Happy New Year”, and all that.

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Some of the philosophers favored asking conjunctive questions, but others argued persuasively that the angel probably wouldn’t count this as a single question. One philosopher wanted to ask “What is the best question to ask?”, in the hope that some day another angel might make a similar offer, at which point they could then ask the best question. But this suggestion was rejected by those who feared that no such opportunity would arise and did not want to waste their only question.

Finally, the philosophers agreed on the following question: “What is the ordered pair whose first member is the best question to ask, and whose second member is the answer to that question?” Satisfied with their decision, the philosophers awaited the angel’s return the next day, whereupon they posed their question. And the angel replied: “It is the ordered pair whose first member is the question you just asked, and whose second member is the answer I am now giving.” And then he disappeared.

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I will write a post soon. Until then, here are some good Epik High songs. For a kid who grew up hating rap, their music sure does resonate with me a lot… I’m realizing more and more that all the things I thought I disliked just needed to be given a chance. I can’t even overstate how much better life becomes when you make an effort to find good qualities about everything, instead of dismissing large amounts of it. Don’t miss out. Good night.

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- Riemann surfaces
- Index theorems (I will have all the prerequisites for this, except algebraic topology)
- Commutative algebra or algebraic topology (which one?)
- 2 CO courses: nonlinear and combinatorial designs (to finish CO major requirements)
- Some lame non-math course (since I need 0.25 more non-math units… sigh)

(If you want to see the official course codes/titles, again, check the “Plan” page). Thoughts?

Here’s a summary of how I’m finding my courses this term. I’m also sitting in on PMATH 955, but I won’t talk about that here. I ordered them by ascending difficulty/time commitment.

**PMATH 445** (Representations of Finite Groups): So far, this course has been really *easy*. I never thought I would use that word to describe a 4th year PMATH course. The assignments are doable in a couple of hours tops, and the material seems pretty standard. Initially I thought it was just because I had experience with representation theory from when I took Lie Groups, but it seems like almost everyone is finding the course easy going. I can’t complain since I honestly don’t think I would have the time to throw at this course if it were much more demanding. It’s my first class at 9:30am, and notes are posted online, so admittedly I haven’t been attending very regularly (things became hectic in the past week or two).

**PMATH 900** (Valued Fields): This course was also easy going in the beginning. It’s a bit more technical now, but still rather palatable since all the objects involved are just fields and certain kinds of rings, and other stuff we all know and love. There’s no highly sophisticated machinery to deal with (I’ll save that for last). The first assignment was very reasonable, but not something I would want to attempt doing in a single day. Algebraic cleverness comes in short bursts for me.

**PMATH 465** (Riemannian Geometry/”Diff Geo 2″): Just like representation theory, the material here is pretty tame, or maybe it just seems that way since (having been through 753, 763, 441, 365, and just about every other course) I’m used to seeing linear algebra everywhere. The concepts seem natural enough to me, although I feel kind of uncomfortable when I have to get my hands dirty and think about solutions of ODEs. I don’t really know the first thing about differential equations. Lectures aren’t hard to follow for the most part, but I just zone out when he starts grinding through horrific tensor calculations, mainly because I have to focus so much on TeXing them, which is always a pain. Assignments are long and routinely absorb my weekends, including the current one.

**CO 430** (Algebraic Enumeration): This is a serious course on enumeration. The operations on species are really cleverly chosen to make the generating functions behave as you’d expect, and they allow you to write down extremely concise formulas that capture some pretty nontrivial ideas. Even if I’m dealing with an equation involving 3 species, it can take me a few minutes to unwrap the definition and grok what it’s actually trying to say. Lectures aren’t the easiest to follow, and the assignments take me a while to solve and write up (he could probably make them significantly more difficult while staying within the margins of reason, though).

**PMATH 822** (Operator Spaces): I don’t even know where to start. A lot of people who took Lie Groups with me last Winter would describe it as the hardest course they ever took. This course is at least 5 times more insane. In addition to the disfigured tensor products everywhere, there are now von Neumann algebras flying around and we never even defined them. The other students (who are mostly PhD students) seem way better than I am at analysis, and math in general. I’m not blaming myself for this, after all I’m just an undergrad; hopefully I will be able to reach a similar level in a couple of years. To be honest, it feels like the lectures are aimed at the PhD students in analysis, and my background in functional analysis just feels inadequate (I also have no measure theory background beyond PMATH 450). Keep in mind that this is coming from someone who generally wrote perfect assignments in Functional Analysis and meticulously checked every detail. It seems like we are also expected to have a lot of time to spend thinking about the material outside of class, which is understandable since grad students usually only take two courses. Unfortunately, I have to deal with five. I can’t even emphasize this enough; if you can’t do functional analysis upside-down blindfolded while reciting the alphabet backwards in your sleep UNDERWATER, you probably barely have a chance of grokking this material during the lecture. I started the first assignment fairly late, and proceeded to sink an enormous amount of time into it. I still didn’t completely finish. I should have known better; operator theory is notoriously subtle. I did feel like I could solve the remaining things with a couple more days, but even so, there was no room in my schedule to continue working on it past today.

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or about the Fibonacci numbers

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, ….

They are intimately related, and I could write several enormous posts enumerating all of their amazing properties.

The golden ratio is an irrational number which satisfies the polynomial equation , that is, we have . In fact, since this is the lowest-degree (hence “simplest”) polynomial annihilating , we refer to it as the *minimal polynomial* of . Many surprising facts can be derived from this innocuous-looking relation. For example, immediately yields that , from which we get the “continued surd” expression

Stated differently, is a fixed point of the mapping . When you were a kid, if you were bored with a calculator, maybe you had the idea of starting with some number, adding 1 to it and taking its square root, and then repeating this process ad nauseum. If you did that, you would have found that eventually the numbers on your calculator stop changing at exactly the value .

For something else, take our original equation , and divide through by to obtain . This tells us is also fixed by the map . It follows immediately that the so-called “continued fraction expansion” of , which (in a precise sense) provides the data of the “best rational approximants” to , must look like:

With a lot of irrational numbers, we get *much* less pretty continued fraction expansions:

One thing to note is that the appearance of a large number in the continued fraction expansion, like 292 above, is telling us something about *Diophantine approximation*: that is, how well we’re able to approximate our number by *rationals of a given denominator*. The rationals you obtain by truncating a number’s continued fraction expansion are provably always the “best” in this sense. Thus, if we look at , whose continued fraction is just all 1’s, we can say that in this precise sense, is the number for which this “approximability” is *the worst*. It is as hostile as can be towards rational numbers.

Anyway, this is just a small taste of . Now let’s do something seemingly random: take the polynomial and “flip” the sequence of coefficients around, to obtain (alternatively, replace each exponent in the expression with the new exponent ). Now we’ll take a reciprocal:

We’re going to look at the series expansion of this thing, which we get from the geometric series formula. For what it’s worth, I don’t care at all about convergence (it’s the summer break; my operator theory course doesn’t start until 2 weeks from now!), so just work completely formally.

Stare at this thing for a while and you notice that the coefficient of in the above is given by

which is (of course) actually a finite sum since we nonchalantly ditch any terms with or . The sequence is the Fibonacci sequence (okay, except for being offset by one or something). It is not hard to show that as , we have . In fact these common ratios are nothing more than the convergents of the continued fraction expansion of .

One cool thing Wikipedia mentions is that you can tile the plane with a “spiral” of squares whose side lengths are given by the Fibonacci sequence:

In the next post I will discuss why all this number-theoretic information related to (root of the polynomial ) shows up in the *power series expansion* of the reciprocal of the “reversed” polynomial . I’ll also apply the same general procedure to , which has the so-called “plastic constant” as its root. The sequence we’ll obtain is called the Padovan sequence, and you can perform a similar tiling of the plane using *triangles* of those side lengths. Once you look at , the sequence you get from the series expansion is no longer nice and monotonic. This is odd, but in hindsight unsurprising since by analogy we would expect it to correspond to some kind of “tiling of the plane by a spiral of 2-sided polygons”, which is absurd.

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This is called *Serre’s affineness criterion*, and the key to the proof (or at least one direction of it) lies in the fact that if you start with an injective -module , and consider its associated sheaf of -modules (just defined by ), then in fact this is *flasque*. We saw before that flasque sheaves are acyclic for the global sections functor , so in particular we can use flasque resolutions to compute cohomology (this will be important later).

We also saw that injective sheaves are flasque, so one might be tempted to claim that the “key” we mentioned above is a mere triviality: indeed, why not just observe that (in view of the equivalence of categories) any injective -module will give rise to an injective sheaf, and then finish? The problem with this argument is that the category of -modules is equivalent to the category of *quasicoherent sheaves*, and **not** the full category of -modules. So yes, we will always have an injective of the former category, but we would need an injective of the latter category to conclude flasqueness — and in general this does not happen.

The starting point is a theorem of Krull from commutative algebra. The full statement concerns the -adic topology on an -module, and I don’t really know much (nor do I currently have time to read Atiyah-Macdonald) about completions. However, we only really need one containment:

**Krull’s Theorem**. Let be a Noetherian ring and be an ideal. If are finitely generated -modules, then for any there is such that .

Now, define the following submodule of :

Before proceeding, let us mention a remark about injectives. We said an object of an abelian category was injective if the functor is exact. This (contravariant) functor is always left exact, so the important thing to take away is the following: “ injective” means that *if is a submodule and is a morphism, then extends to a morphism *.

Surprisingly, the above turns out to be equivalent to the following seemingly weaker condition (*Baer’s criterion*), namely: if is an ideal of and is a morphism, then extends to a morphism . This equivalence is a basic result from commutative algebra.

This reminds me of a similar thing that came up when trying to formulate the universal property of the Stone-Cech compactification: in some sense the closed unit interval is a “good enough” representative of the class of *all* compact Hausdorff spaces (this is formalised in the fascinating notion of an *injective cogenerator*).

**Lemma 1**. Let be a Noetherian ring, let be an ideal. Then if is an injective -module, then is also an injective -module.

To prove this, we only need to establish Baer’s criterion for , and this is done by observing one can apply Krull’s theorem to the inclusion , pulling back from to , and finally using the natural map to pull back to as required.

**Lemma 2**. Let be a Noetherian ring, and an injective -module. Then for any , the natural map to the localisation is surjective.

This lemma isn’t very difficult either. If is defined as the annihilator of , then you get some ascending chain of ideals in , but is Noetherian, so , yada yada. Then, letting be the natural map, you take some , write for some and (you can do this by definition of localisation), and define a map by sending (this turns out to be fine since as -modules, and ). Lift to a map by injectivity of , and then let . Then . Magic.

**Proposition**. If is an injective -module, then is a flasque sheaf of -modules, where .

To establish this, we use Noetherian induction on the support of the sheaf (call it ). The basic idea is, for some open , to choose some and consider some open of the form . Noting that , we can invoke the lemma above, and then the problem reduces to showing is surjective, where . But this follows by induction (put and note this is an injective -module by the lemma, hence , whose support is strictly contained in , is flasque; at this point we win since for all opens ).

**Theorem**. Let for some Noetherian ring . Then if , for all quasicoherent sheaves on .

To see this, let . Take an injective resolution in the category of -modules, apply the Serre functor to get a flasque resolution of . Applying the global sections functor, we just get back the original resolution, so we’re done.

**Theorem** (Serre). Let be a Noetherian scheme. Then TFAE:

- is affine.
- for any quasicoherent sheaf on and .
- for any coherent sheaf of ideals on .

We’ve already shown that (1) => (2), and (2) => (3) is easy. (3) => (1) can be proved using the following characterisation of affineness: is affine if and only if there are such that , each set is affine, and is covered by .

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R: What is

L:

R: What is the quantum analogue of

L: I have no clue.

R: Let’s try This is the sequence

L: OK. . I can’t get anywhere from here that would make it look remotely close to .

R: What do we do?

L: We call upon the fire of inspiration itself, Ignis.

IGNIS: The truth ye seek is but a cascading staircase, but one that ends.

L: What do you think that means?

R: Instead of trying , let’s try

there are terms in total just like , but here, each term is one less than the previous.

L: Like a factorial that suddenly ends. All right, so

Oh! The factors line up. The subtraction yields

This is exactly what we are looking for.

R: Great! So we have the identity

corresponding to

in the real calculus.

L: OK. What about when or when ?

R: Well… what do we do to get from to

L: Well, we divide by its last term, which is , to get to , because the only difference between them is that is missing the term.

R: What do we get when we go from to

L: Well, , and so we divide this by to get to , so I guess we should define

R: What about

L: Well, we would divide this by so I guess it is

R: So what’s the general formula?

L:

Are you sure the identity (*) still holds for these new definitions?

R: For these situations, we should call upon the minion of non-illuminating bashing, Grunt.

GRUNT: ‘Tis done.

R: There, this fact has been verified.

L: I shall call these Pochhammer symbols.

R: WTF?

L: Or we can call them falling factorials. Or falling powers.

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