nerdifying the world

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.

Adeel Khan has set up a git server for the CRing project. In particular, you can follow how open source commutative algebra evolves in real time. More practically, you can download source files from there; they’re also on the main website, of course, but the ones there are likely to be slightly newer: I can’t update the website on my college account instantly. Plus, you can see who contributed what in what is really an intuitive and transparent manner. You can also use the git server to submit contributions, though for that you’ll need the password. For this, you can write to cring.project(at)gmail.  Again, if you don’t want to use git, we’re happy to receive contributions by email.

So why and how should you contribute? Johan deJong has explained it here (for the Stacks Project); the same applies to the CRing project. After all, the source code to your old homework sets or class notes isn’t doing anything on your hard drive. We’d be thrilled to receive it and to list you as a contributor. And we’ll work out how to edit it in (unless you want to, which you are welcome to).

[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…)

There is a new Q&A website, currently just called math.SE, for discussion of mathematics at any level. Part of the point is to provide a useful complement to MathOverflow at a less advanced level.

The website has just moved into public beta, and ideally will become a full-fledged StackExchange website in a few months. I’d encourage you to give it a try! It’s not quite at the same level as MathOverflow, admittedly, but even if that bothers you, you’ll at least get to see what SE 2.0 (which I’ve heard MO is going to migrate to) is like.

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.

Remember the Sokal affair?  That was when an NYU physics professor submitted a parody article ostensibly about science, but using meaningless jargon to a journal of cultural studies, and it got accepted.  Oops.

Well, David Simmons-Duffin, a graduate student in theoretical physics at Harvard, has created a similar parody site called the snarXiv.  So far, the site uses context-free grammars to randomly generate meaningless abstracts involving fancy terminology.  For instance,

We verify an involved correspondence between decay constants in supergravity deformed by multi-fermion operators and path integrals in superconformal superconformal QFTs surrounded by (p,q) instantons. The determination of superconformal effects localizes to AdS_n x P^m. Therefore, some work was done among mathematicians on a model of bubbles. This result has long been understood in terms of the Wilsonian effective action. The Virosoro algebra is also bounded. After reviewing fragmentation functions, we derive that spinodal inflation at $\Lambda_{QCD}$ depends on the Seiberg-dual of the Landau-Ginzburg Model.

There is also a game where you can try to distinguish the fake abstracts from the real ones (on the arXiv, the actual site). I’m ashamed to say that I’m worse than a monkey at physics.

Now, someone who knows about programming should do the same for mathematics, and use the key words: “moduli spaces, etale cohomology, $latex \infinity$-groupoidification, Deligne-Mumford stacks, perverse sheaves, Calabi-Yau manifolds, and homotopical category theory.”

Edit: Wait, there’s more!  Apparently, the creator has a theorem generator and even a program that can generate philosophy.

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.

I have a math blog? What is this?

The main excuse I had for ignoring Mount Bourbaki for the past month or so was the Intel science competition, which ended last week.  It was a lot of fun—I met many interesting people and enjoyed numerous pleasant conversations.

To my surprise, I ended up coming in third place.  I was quite stunned by this especially after hearing the finalists called before me–I have to say that I was genuinely amazed by every project that I saw.

Of course, I can’t resist a picture.   Here is one from the gala, of the top three:

I’m the guy on the left looking in the wrong direction.

I probably will do a technical post at some point about what my project was all about, but for now here is a non-technical video I made:

Other than this, I also know now what I’m doing this summer.  I’m going to do an REU at Penn State, where I’ll be working on a topic that I should probably find out about soon.   What I’m going to do next fall is still undetermined.

As of late, I’ve been reading up on some logic and model theory and a bit of ergodic theory (from Walters’ book, which I recommend).  I tried to study Spanier while I was at Intel though didn’t get very far.  And I sincerely will try to get some entries up soon.  I don’t know whether I will be able to keep my promise of Grothendieck topologies just yet.

The main reason I’ve been neglecting this blog for the past few weeks is to prepare for the Intel STS, a science fair for high school seniors, where I’m currently at to present my RSI project on representation theory in complex rank.  I’m going to experiment in liveblogging.  Except it really belongs over at Delta Epsilons, so you’ll find it there.

The book Real and Complex Analysis, by Christopher Apelian and Steve Surace, was recently released.

It’s mainly for an introductory upper-level undergraduate course in real and complex analysis, especially at small liberal-arts colleges.  In this post, I’ll describe this book and how I was involved in its production.

At the start of my freshman year, my analysis teacher, Professor Surace, asked me to check over the drafts of a book he and his colleague (and my former teacher) Prof. Apelian were working on. It was the textbook for the course. At the time, if I remember correctly, there were six chapters: on the real and complex spaces, basic topology, limits, continuity, convergence of functions, and derivatives. The complex analysis part of the book was in its infancy (e.g., there was only a rudimentary outline of one chapter, which had been written some time back and was typeset in Word—they wrote it well before before they had switched to LaTeX).

Anyway, since one of my hobbies that year was playing around with LaTeX and trying to figure out all the cool formatting tricks used in books, and since they hadn’t yet done much to change the (somewhat bland) default book style, I pointed out a few tweaks that, in my opinion, would make it look better. They agreed, and I ended up being put in charge of the layout. It turned out that I would need to learn a lot more about LaTeX though (or at least, learn to look up things a lot more). As you can see from the sample pages, the authors—or to be precise, one of them :-)—had fairly detailed ideas of how the book should appear. I don’t know if I learned LaTeX properly, but I sure learned a lot of hacks.

I also ended up sketching the figures, which enabled me to pick up the useful (and definitely nontrivial) skill of using Adobe Illustrator. In fact, I’ve used it in making some of these blog posts.  (Though I would recommend interested passers-by also to consider, say, Tikz depending on your aims; I’m not familiar with it, but it has the benefit of being free.) (more…)

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