Alexandre Borovik writes about a disturbing story coming out of Turkey. Ali Nesin organizes a summer school in mathematics in a village near Epheseus in Turkey. For some sort of bureaucratic rule violations (including “education without permission”), he has been arrested and the school has been shut down.
Alexandre has a follow-up post with more details, and a online petition. Contact details are provided if you want to add your name to the petition.
I was thinking about nets the other day, when I was reminded of something that I wondered when I first encountered them. Is the generalization to partially-ordered sets strictly necessary? The generalization to partially-ordered sets is useful, but are there spaces with points that are not reachable by totally-ordered nets alone?
In a first countable space, a point lies in the closure of a set if and only if there is a sequence of points in the set that converges to the point. This property fails in general. For example, an uncountable set with the cofinite topology has no non-trivial convergent sequences, but the closure of any infinite set is the whole space. You can recover this property if you pass to nets, which allow fairly general partially-ordered sets to be the index set (the only requirement you must impose is that they be directed sets). So if you require the index sets of your nets to be totally ordered, is there a space which contains a point that is not the limit of such a net?
Poking around Wikipedia, I found that the page for order topology, which suggests that the Tychonoff plank is an example of a space where totally-ordered nets are not sufficient. The page discusses nets indexed by ordinals, which possibly is a loophole, but it seems like a very narrow one. I’d be curious if anyone knows for sure.
Actress Danica McKellar, who co-authored a paper in mathematical physics as an undergraduate, has now written a book, Math Doesn’t Suck: How to Survive Middle-School Math Without Losing Your Mind or Breaking a Nail. The book is aimed at girls in middle school who might be scared away by math.
Via Aetiology, who has a more detailed review of the book.
In a post on his weblog, Michael Greinecker mentioned some applications of homology to economics. While aimlessly websurfing for more information, I came across the homepage of Graciela Chichilnisky, a mathematical economist who has extensively applied topological techniques to economic questions. Chichilnisky has written nearly 200 articles, and includes PDFs on her site.
On a less happy note, Chichilnisky also links to a site dedicated to detailing her problems with Columbia University, where she is a tenured professor. Chichilnisky had successfully sued Columbia on the grounds of sex discrimination in the 90s. The two parties are now back in court over an alleged pattern of retaliation on the part of Columbia. An article on her experiences with the reviewing process also makes depressing reading.
There are two new graduate-level math weblogs out there. The oddly-named Secret Blogging Seminar and modestly-named The Everything Seminar both cover a broad range of advanced topics. Enjoy them while you can — as you can see here open hostilities between the two are about to break out.
I have long been a fan of John Cramer’s Transactional Interpretation of Quantum Mechanics [Wikipedia link here.], mainly because it appeals to my “Trust the math” outlook towards physics models (which in turn probably goes a long way in explaining why I am not a physicist).
Apparently his proposed experiment is in the news again and I can’t seem to find any real info about the current state of affairs/partial results. Does anyone have know anything more up to date?
The twelfth Carnival of Mathematics is up at The Vedic Maths Forum India Blog.
This is a little late, but the eleventh Carnival of Mathematics is up at Grey Matters.
In comments, Grétar Amazeen asks:
Is Langs Algebra a good book? I just got it in the mail and I´m going to use it to brush up on my algebra before I start graduate school. I´ve heard that he uses his own private nomenclature, is that something I´ll have any problems with?
Michael has already given word-for-word my answer to the question:
Oh dear god no.
Lang does use his own private nomenclature (“entire rings”, for example), but that’s a minor issue. The book is just hard to read. The only chapter that I thought was well-written was the group theory chapter, but it’s very concise, so it might not be good for your purposes.
Abstract algebra has two excellent textbooks that are pitched at the advanced undergraduate or introductory graduate level: I. N. Herstein’s Topics in Algebra, and Michael Artin’s Algebra. Herstein covers the standard topics very clearly. Artin gives a much broader introduction to algebra’s relationship to other fields of mathematics, so it’s good for inspiration.
A few topics not covered in Herstein that are worth knowing are:
(These are probably all covered in Artin, but I don’t have my copy handy so I’m not completely sure. They are all covered in Lang, but in both cases the chapters aren’t very good.)
A more idiosyncratic suggestion I have is Ideals, Varieties, and Algorithms, by Cox, Little, and O’Shea. It covers the Nullstellensatz, but from the point of view of Gröbner bases, which are a computational tool that makes it easy to work out examples in commutative algebra. They also make it easier to understand why homological algebra is interesting from an algebraic point of view, and not just as a tool in algebraic topology, again because they make examples easy to work out.
Guest posting at the n-Category Café, Tom Leinster drew everyone’s attention to Sheldon Axler’s manifesto, Down With Determinants! Axler argues when teaching linear algebra, determinants should be avoided almost completely.
This has led to a surprisingly wide-ranging discussion. Lieven Le Bruyn and more recently Masoud Khalkhali have weighed in with defences of the determinant.