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Real and Complex Analysis, by C. Apelian and S. Surace, now published December 19, 2009

Posted by Akhil Mathew in analysis, General, math education.
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(This material is reposted from here.)

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). (more…)

Two massively collaborative mathematical websites that readers may like October 17, 2009

Posted by Akhil Mathew in General, math education.
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I realize that I’m late to the party on this, but there is a new mathematical website precisely for answering questions: Math Overflow.   Modeled on the Stack Overflow site for programmers, Math Overflow seems to have done a nice job in attracting a large crowd of professional mathematicians and students.  Questions tend to be answered quickly, and there is an interesting “reputation” feature that measures one’s respect in the community.  This is probably a much better approach than tossing out blegs (for readers here that are bloggers) since many more people will read it, and since the questions will be available in a common source for other mathematicians.  Since there are other sites that have active communities for math help, Math Overflow restricts itself to questions that are “of interest to at least one mathematician.”   (more…)

Open source textbooks October 10, 2009

Posted by Akhil Mathew in General, math education.
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Well, it seems that the Bourbaki 2.0 idea I suggested some time back wasn’t entirely absurd: as a commenter pointed out, the Stacks Project is following a similar model.  Moreover, Nathan Dunfield of Low Dimensional Topology has proposed that the stacks model be applied to textbooks (I assume the stacks book is more of a reference).  Additionally, he asks why conventional textbook publishing, even for individual authors, is still necessary in the day of the internet when it is more efficient to distribute material online.   Some people have apparently listened to these ideas; Jacob Lurie, for instance, has put his treatise on higher topos theory on the arXiv, and Allen Hatcher has made available his well-known text on algebraic topology on his webpage

I’d very much like to see this trend continue; there are surely people out there who would like to learn mathematics beyond the introductory calculus and linear algebra level–when there are no longer massive surpluses of texts on one topic–but may not be affiliated with a university for various reasons, and may not want to fork over the substantial sums that conventionally printed textbooks cost these days.  At least for authors, I don’t think there’s much money to be made in algebraic topology writing, and math professors have nice salaries anyway, so why not?

A talk on the p-adic numbers September 16, 2009

Posted by Akhil Mathew in algebraic number theory, General, math education, number theory.
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The start of the academic year has made it much more difficult for me to get in serious posts as of late, and the number theory series has slowed.  Things should clear up at least somewhat in a few more weeks.   In the meantime, I’ll do something that occurred to me a while back but I then forgot about: posting a talk.

I took an independent study course last semester on class field theory.  As is traditional, I gave a talk last May after the course on some aspects of the subject matter.  Several faculty members at the university and teachers in my school attended, along with some undergraduates there.  In the talk, I gave an elementary overview of the p-adic numbers, assuming no more than basic number theory and point-set topology.

Anyway, I am posting the (slightly corrected) presentation and the notes here.

Bourbaki 2.0: Or, is massively collaborative mathematical exposition possible? September 7, 2009

Posted by Akhil Mathew in General, math education.
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Warning: I have very little knowledge about these topics (even less than usual).

The Problem

One of my goals is to learn mathematics independently. I’ve had lots of trouble especially in certain areas such as algebraic geometry, where the preqrequisites are large and interconnected. When reading books nowadays, I frequently come across words I don’t know with (sometimes) recommended supplementary sources. But I can’t really learn the definition of say, a Cohen-Macaulay ring, just from reading Hartshorne’s short blurb or Wikipedia without actually seeing some properties of these rings proved, so I go to the supplementary sources. When I looked up, say, Matsumura’s book on commutative algebra, I then find that I am expected to know what derived functors are to understand depth. Time to find another book!  (more…)

More math illiteracy August 20, 2009

Posted by lumixedia in General, math education, number theory.
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Just for fun, here’s a rather pointless anecdote.

My third-grade teacher decided to have a fun, hands-on activity to teach our class about primes. Now I have a low opinion of all fun, hands-on activities (give me a good, proper whiteboard lecture any day, and if you’re incapable of doing so you should really just work on improving your teaching skills before making me pay attention to you, and yes, this was my opinion even when I was very, very young) but that’s not the point of this post. (more…)

Bleg: Mathematics textbooks for high school self-study August 16, 2009

Posted by Akhil Mathew in blegs, General, math education.
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Inspired by Martin’s latest entry, and the general difficulty of teaching oneself mathematics, I’m going to shamelessly copy imitate the the idea of a recent Ben Webster post about undergraduate summer reading to ask our readers:

What books would you recommend for self-study by a serious high school student interested in mathematics, physics, or computer science?


Another math illiteracy moment August 15, 2009

Posted by lumixedia in General, history of mathematics, math education, number theory.
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I was recently informed that the Goldbach conjecture is popularly known in China as the “1+1=2” conjecture. As in, “every positive even number can be written as the sum of two primes. For example, 1+1=2.” [Edit–I was told this by a Chinese person who might nevertheless not be representative of how this nickname is understood–see comments.]

When I mentioned that this nickname is not in fact accurate, the person who so informed me got rather annoyed with my pointless pedantry. Why shouldn’t 1 be prime? Why not define a “prime” to be a positive integer with at most two distinct divisors, rather than a positive integer with exactly two distinct divisors? Clearly the “1+1=2” conjecture sounds way cooler than the “2+2=4” conjecture to a layman, and we are talking about popular mathematics here, so why not?

Okay, I guess it might not be immediately obvious why current notation is preferable. Maybe. From a certain perspective. It is also admittedly true, according to Wikipedia, that 1 was indeed widely considered to be prime by mathematicians up to a few hundred years ago. Fine. So let’s temporarily redefine “prime” to mean a positive integer with at most two distinct divisors, and see if it’s acceptable today. (more…)