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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?

Riemann integration in abstract spaces September 30, 2009

Posted by Akhil Mathew in analysis.
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I’ve been busy as of late with college applications and a science competition. But now I have a bit more time, so I shall try to resume posting.

Anyway, speaking of science competitions, I participated in the Intel International Science and Engineering Fair in 2007 with a self-guided project. The bulk of it dealt with Riemann integration in abstract spaces and the potential for generalizing certain constructions in analysis to this setting.

After the competition, I tried submitting a condensed version of the material to a mathematical journal, which concluded that the work did not merit publication, but may have had some interest: the method, while contained in more general approaches, seemed to have not been taken in the literature. (Unfortunately, I was unaware of the literature.) 

The paper I submitted is here.

Nevertheless, since this is not a professional blog, I thought this might be an appropriate setting to post the paper and briefly discuss it, so I will try and see how this goes. (more…)

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…)