math education

I’ve been away from this blog for longer than I should have. I got stuck in my series on the cotangent complex, partially because I’ve been busy doing other things–namely, trying to learn about the foundations of etale cohomology. As I learn more I might write a few posts. And someday the cotangent complex thing will get finished as a short expository note on my website.

One thing I’ve discovered as of late is that many concepts that I learned earlier in life were in fact shadows or special cases of more powerful and general ones. I’ve consequently had to un-learn many such concepts, to replace them with the newer ones.

Sheaves

An example is basic sheaf theory: like many people, I learned this from Hartshorne chapter II, working out the exercises there. But as I have more recently discovered, many of the methods there are not the appropriate ones for the general theory of sheaves on a site. As an example, Hartshorne defines sheafification (and many other things) on a topological space using stalks. However, on a site this is meaningless because there is no analogous notion in general.

The stalk of a sheaf (or presheaf) on a space $X$ at a point corresponds to the inverse image functor via the inclusion $\{\ast\} \to X$. The analogy in the theory of sites would be the inverse image via a morphism from the site with one point (or something equivalent to this). It turns out, fortunately, in etale cohomology this more general notion does make sense, if $\{\ast\}$ is taken to be the spectrum of a separably closed field. So, if $X$ is a scheme, it is not topological points $\{\ast \} \to X$ that lead to the stalk functors in etale cohomology, but the morphisms $\mathrm{Spec} K \to X$ for $K$ a separably closed field (e.g. the separable closure of the residue fields of the topological points).

It is a curious story that there is an even more general theory of points of a (Grothendieck) topos. A point is a geometric morphism (that is, an adjunction where the left adjoint is exact) between the category of sets and the given topos. The direct and inverse image functors obtained from maps $\mathrm{Spec} K \to X$ show that there are lots of “points” in the etale topos. In fact, on general so-called “coherent” topoi there is a general theorem of Deligne that there are always enough points to detect isomorphisms of sheaves. Apparently this is a topos-theoretic reformulation of the completeness theorem in first-order logic! I’m far from understanding the story here though. (more…)

The CRing project now has a blog. As a result, I’ll be able to return to normal posting here, while discussion about CRing will be able to occur freely there.

[The present post is an announcement of the CRing project, whose official webpage is here.]

Like most mathematics students, I spend a lot of time writing stuff, for instance homework assignments and (of course) blog posts. So I have a lot of random, unorganized write-ups littered around my hard drive, which might be useful to others if organized properly, but which currently slumber idly.

Last semester, I took a fairly large amount of notes for my commutative algebra class (about 160 pages). I made the notes available on my webpage, and was pleased with the reception that they received from my classmates. After seeing Theo-Johnson Freyd’s projects, I decided that it might be a productive exercise to edit the notes I had taken into a mini-textbook. I quickly made progress, since the basic structure of the book was already set by the lectures. I decided early on that the work was going to be open source: to me, it seemed the best way to ensure that anyone who wanted could freely access and modify it.

But I think the project is bigger now. Namely, instead of an open source textbook, I want a massively collaborative open source textbook. This is to say that I don’t want it to be my work anymore, but my work as well as, and more importantly, the work of enthusiastic professors, procrastinating graduate students, nerdy high-schoolers,  or whoever else wishes to contribute. The goal is to end with an openly available textbook suitable for a beginner familiar only with elementary abstract algebra, but which will provide adequate preparation for the serious study of algebraic geometry.

So, I present you the CRing project. (more…)

So I signed up to give a talk at HMMT some time back, which will be this Saturday. As expected, I procrastinated preparation for it until now. The problem is, I’m not sure what to talk about. In high school, I wasn’t really into math contests such as HMMT — my mind was never able to find creative solutions with the necessary speed, and I’d consistently turn in abysmal performances. So as a result, I was never exposed to much of the culture of high school math contests (the existence of which I found out not that long ago). Anyway, as a result I’m not completely sure how to prep this talk, or even what to talk about. Some topics that I consider talk-worthy and interesting are:

1. Lecture one of algebraic geometry class. Define varieties and algebraic sets, and state (or even prove) the Nullstellensatz. But I suspect this will use too much commutative algebra than I should assume. I understand that plenty of extremely accomplished HMMTers may not know what a ring is.
2. The p-adic numbers.  This has the benefit of my being able to recycle an old talk.  But I might have to re-tool it.
3. Quadratic reciprocity. Perhaps the proof via Gauss sums, for instance. But this is something that people will tend to know, right?
4. A brief intro to computability theory (as in — Turing machines, unsolvability of the halting problem, complexity classes, maybe say something about Kolmogorov complexity)

The basic problem is that such topics essentially amount to picking your favorite textbook on subject X, choosing five or six pages, and reading them aloud to the students — in short, a normal class. Which is probably not what they’re looking for.

But some of you readers have better ideas than I.  So, any thoughts? Pretend, or not, that you were in high school. What would you wish to know that I could cover in an hour?

(If I end up using your topic, I’ll mention you in the talk!)

There is a new program called PRIMES for high school students in the Boston area. There are lots of high school math programs out there, but this one is fairly unique in offering a research experience. (RSI is the only one I know of, and PRIMES is based on it.) Also, unlike most of the others, it is during the school year as well–that’s why it’s only for kids in Boston.

Anyway, it looks like a very nice idea, and the mentor-in-charge, Pavel Etingof, has a fair bit of experience with this sort of thing. So if you’re a high school student in the area, apply! Or if you know one, tell them about it.

Here at PSU, Sergei Tabachnikov just finished giving a two-week mini-course on the “geometry of polynomials.” The collection of topics was diverse: various proofs of the fundamental theorem of algebra, resultant and discriminants, Chebyshev polynomials, harmonic functions in three-space, and a sketch of the proof of the four-vertex theorem. The lectures presupposed familiarity with no more than elementary analysis and linear algebra, though more advanced topics were referenced (without proofs).

Two weeks really means about seven days; class was cancelled because apparently people thought that most of us were interested in an art show.

For the benefit of the huddled masses yearning to be educated, here are the notes that I took from these lectures.  The file is rather large (40 MB) because of the insertion of jpg images that someone else in the REU drew. The djvu file is a lot smaller, but WordPress won’t let me post it, so email me if you want it.

The notes are mostly a faithful representation of what I took in class, but I have edited them lightly to moderate my tendency to embarrass myself.

Next week, Yakov Pesin is lecturing on fractal geometry and dynamics; I’ll post those notes when I’m done with them.

Tim Gowers asked a really great question on MathOverflow recently, on examples of mathematical “cognitive biases”: false widely held beliefs in (higher) mathematics.  Which naturally enough reminded me of the embarrassing experience yesterday when I realized that, after assuming the contrary for several months, the kernel of $A \oplus B \to C$ is not the same as the direct sum of the kernels of $A \to C$ and $B \to C$. Whoops.

It looks like the winning ones so far are about little facts in linear algebra.  It would, indeed, make the proofs of many of the technical results on Lie algebras easier if $\mathrm{Tr}(ABC) = \mathrm{Tr}(CBA)$.

Also, Arrow’s theorem is a scam.  Support range voting!

Finally, on an entirely unrelated note, this quote is ridiculously awesome:

“I’ve had the chance, in the world of mathematics, to meet quite a number of people, both among my elders and amoung young people in my general age group, who were much more brilliant, much more “gifted” than I was. I admired the facility with which they picked up, as if at play, new ideas, juggling them as if familiar with them from the cradle — while for myself I felt clumsy, even oafish, wandering painfully up an arduous track, like a dumb ox faced with an amorphous mountain of things I had to learn (so I was assured), things I felt incapable of understanding the essentials or following through to the end. Indeed, there was little about me that identified the kind of bright student who wins at prestigious competitions or assimilates, almost by sleight of hand, the most forbidding subjects.
In fact, most of these comrades who I gauged to be more brilliant than I have gone on to become distinguished mathematicians. Still, from the perspective of thirty or thirty-five years, I can state that their imprint upon the mathematics of our time has not been very profound. They’ve all done things, often beautiful things, in a context that was already set out before them, which they had no inclination to disturb. Without being aware of it, they’ve remained prisoners of those invisible and despotic circles which delimit the universe of a certain milieu in a given era. To have broken these bounds they would have had to rediscover in themselves that capability which was their birthright, as it was mine: the capacity to be alone.”  -Alexandre Grothendieck

Source.

A friend of mine is taking a course on analytic number theory in the spring and needs to learn basic complex analysis in a couple of weeks.  I decided to do a post (self-contained, except for Stokes’ formula) on deducing the Cauchy theorems and their applications from Stokes’ theorem now instead of later–when I’ll talk about several complex variables.  It might be objected that Stokes’ theorem is just Green’s theorem for $n=2$, commonly used in undergraduate treatments, but my goal was to take an expository challenge: write something rigorous on complex variables in as short a space as possible without sacrificing readability.  So Stokes’ theorem for manifolds is preferable to Green’s theorem as stated in a vague way about “insides of a curve” (before, say, the Jordan curve theorem is proved) and the traditional proof of Green’s theorem via rectangular decompositions.

So, let’s consider an open set ${O \subset \mathbb{C}}$, and a ${C^2}$ function ${f: O \rightarrow \mathbb{C}}$. We can consider the differential

$\displaystyle df := f_x dx + f_y dy$

which is a complex-valued 1-form on ${O}$. It is also convenient to write the differential using the ${z}$ and ${\bar{z}}$-derivatives I talked about earlier, i.e.

$\displaystyle f_z := \frac{1}{2}\left( \frac{\partial}{\partial x} - i \frac{\partial}{\partial y} \right) f, \quad f_{\bar{z}} := \frac{1}{2}\left( \frac{\partial}{\partial x} + i \frac{\partial}{\partial y} \right) f.$

The reason these are important is that if ${w_0 \in O}$, we can choose ${A,B \in \mathbb{C}}$ with

$\displaystyle f(w_0+h) = f(w_0) + Ah + B \bar{h} + o(|h|), \ h \in \mathbb{C}$

by differentiability, and it is easy to check that ${A=f_z(w_0), B=f_{\bar{z}}(w_0)}$. So we can define a function ${f}$ to be holomorphic if it satisfies the differential equation

$\displaystyle f_{\bar{z}} = 0,$

which is equivalent to being able to write

$\displaystyle f(w_0 + h) =f(w_0) + Ah + o(|h|)$

for each ${w_0 \in O}$ and a suitable ${A \in \mathbb{C}}$. In particular, it is equivalent to a difference quotient definition. The derivative ${f_z}$ of a holomorphic function thus satisfies all the usual algebraic rules, under which holomorphic functions are closed. (more…)

I’ve created two new pages: a bibliography, and a collection of old writings.

The bibliography is for me to list the sources I use (or plan to use shortly) in writing this blog.  It’s also a list of books that I’ve found helpful in various fields, so perhaps it will be useful to others learning about the same type of material.  Currently I’ve found the books by Folland, Introduction to Partial Differential Equations, and Taylor, Partial Differential Equations, to be excellently written and well motivated. Taylor’s book also ties in the differential geometry (more so than Folland).   The style is uniformly clean but leisurely, and both books are more-or-less self-contained.

Also, I collected together some old notes I’ve written for various reasons and posted them in case anyone might find them helpful.  For instance, I wrote an expository paper on integral equations a few years back for a seminar course, which I was pleased to stumble across, since I did not remember it.  That may become a blog post in the future.

While reading Feynman’s autobiography a few days back, I came across a funny story. As an MIT undergraduate, he explained to a fellow student in a drawing course a “special” property of the French curves they used: “At the lowest point on each turn, no matter how you turn it, the tangent is horizontal. His fellow students thought apparently thought this was remarkable and spent some time playing around with the curve, despite having already taken calculus.

I ran into the same situation today. While working out an exercise on an ordinary differential equation in the plane, I wondered if a smooth curve ${c: I \rightarrow \mathbb{R}^2}$ whose tangent vector ${\dot{c}(t)}$ is always pointing in the same direction as ${c(t)}$ lies on a straight line through the origin. A reparametrization by arc length allowed me to figure it out quickly.

A few minutes later, though, I realized something. The condition above was just that the tangent vectors of ${c}$ are parallel—so ${c}$ is then a geodesic with respect to the usual connection on ${\mathbb{R}^2}$, up to reparametrization! And geodesics in this case are just straight lines.  So it should have been immediately obvious to someone who had spent a month blogging about nothing (and immersed in) but differential geometry.

This was probably a bad example, but I think the point stands that perhaps books ought to emphasize these kinds of little insights more to avoid creating this kind of fragility. Amusingly, Feynman says he found a similar problem (albeit obviously with more complicated material) when a graduate student at Princeton talking to people who worked there.

Next time I will go back to actually blogging about mathematics.

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