Showing posts with label Terence Tao. Show all posts
Showing posts with label Terence Tao. Show all posts
10 October 2008
20 November 2007
the Tao of compactness
Terence Tao on compactness and compactification (from the forthcoming Princeton Companion to Mathematics). Particularly interesting is the fact that using ideas of compactification, one can rigorously view a continuous object as a limit of discrete objects: the circle group as a limit of cyclic groups, straight lines as the limit of circles (they're just circles with the center at infinity!), the Dirac delta function as the limit of taller and narrower "spike functions", and so on.
Since I seem to be on a nomenclature kick, "compact" is kind of misleading nomenclature at first; if you didn't know what "compact" meant, wouldn't you think that if the closed interval [0, 1] is compact then the open interval (0, 1) surely must be? After all, "compact" means something like "small" in everyday discourse, and (0, 1) is contained in [0, 1]. It seems even sillier when we say that the real line is compactified by adding a "point at infinity"... until you realize that this means you can just join the edges together and turn the line into a circle, and circles are "smaller" than lines. Intuitively "compact" should mean what "bounded" means, and of course the Heine-Borel theorem says that in Euclidean space "compact" means "closed and bounded". Why should something called "compact" and something called "closed" be connected? And why aren't "closed" and "open" logically opposite? When I first learned the definitions of those -- the definitions on the real line, in a real analysis class -- I must have flipped back to the page with those definitions on them in baby Rudin at least once a day. I eventually made my peace with those definitions... but it took a while.
Since I seem to be on a nomenclature kick, "compact" is kind of misleading nomenclature at first; if you didn't know what "compact" meant, wouldn't you think that if the closed interval [0, 1] is compact then the open interval (0, 1) surely must be? After all, "compact" means something like "small" in everyday discourse, and (0, 1) is contained in [0, 1]. It seems even sillier when we say that the real line is compactified by adding a "point at infinity"... until you realize that this means you can just join the edges together and turn the line into a circle, and circles are "smaller" than lines. Intuitively "compact" should mean what "bounded" means, and of course the Heine-Borel theorem says that in Euclidean space "compact" means "closed and bounded". Why should something called "compact" and something called "closed" be connected? And why aren't "closed" and "open" logically opposite? When I first learned the definitions of those -- the definitions on the real line, in a real analysis class -- I must have flipped back to the page with those definitions on them in baby Rudin at least once a day. I eventually made my peace with those definitions... but it took a while.
21 August 2007
links for 21 August
Terence Tao writes about the theorem that Danica McKellar, of Wonder Years fame, proved; she's been in the news a lot lately because of her book Math Doesn't Suck: How to Survive Middle-School Math Without Losing Your Mind or Breaking a Nail
. A large part of the mainstream media said "she even proved a theorem! and it was published!" but nobody explained what the theorem was, because it takes a few pages of exposition. But Tao does a good job with it, getting in some nice statistical mechanics on the way.
(The paper itself, "“Percolation and Gibbs states multiplicity for ferromagnetic Ashkin-Teller models on Z2” " is available at McKellar's web site; I suspect this is the only instance of a mathematical theorem on a celebrity's web site, at least for suitable definitions of "celebrity". The theorem, by the way, says that the Curie temperature and the critical temperature for site percolation are equal in the Ashkin-Teller model on the square lattice.
Scott Morrison at Secret Blogging Seminar (which is becoming less secret every day) gives us grepping the Knot Atlas, which is about the Knot Atlas. The Knot Atlas is basically Wikipedia for knots. (The Knot Atlas, by the way, has no entry called "knot" or something similar that I could find. It doesn't define knots, it only tabulates them.) A knot is not exactly what the layperson would think it is; a mathematical knot is an embedding of a circle in 3-space, so the string has no ends. (Of course, one can represent any knot tied with an actual rope as a mathematical knot by splicing the ends together.) This is a continuation of the old Knot Atlas; it's a lot easier for anyone to edit a wiki. It is simultaneously a database of knot invariants, and the post I linked to is about how to extract that data.
One of the things that has always amused me about knot theory is Lord Kelvin's theory that atoms were knots; but psuedoscience often begets science. I once worked at the Department of Alchemy.
Another such wiki is qwiki for quantum physics. I don't know of any more but they seem like they could be useful, as references to various mathematical areas built by the community that actually uses them. Wikipedia plays this role to some extent, but its main function is to be a general-purpose encyclopedia so the technical details often don't make it in.
I find myself wondering if people have used wikis (with login, of course) to write mathematical papers with collaborators. (Or even alone! Since most wiki software stores old versions of pages, you could easily throw out arguments that you didn't like, confident that if they contained some seed of truth they could be recovered later.)
At Casting Out Nines, via Vlorbik: inside facts about calculators. The big one is that nobody in the "real world" actually needs a calculator; students generally have computers now, and the student versions of Maple and Mathematica are similarly priced to graphing calculators; therefore if it makes sense for students to use any sort of calculator, it should be what's usually called a "scientific" calculator. I agree. However, I have a ten-year-old TI-83 which I know pretty well, and I generally keep it around -- for when I'll be working in a coffee shop and don't feel like taking my laptop with me, or for when my officemate is using the computer. But unquestioning faith in the calculator is a bad thing. You can also read Eric Schechter's The Most Common Errors in Undergraduate Mathematics. Would it be possible to compile an analogous list for graduate students, or research mathematicians? Or do those of us at these levels make too many different errors for this to be possible?
. A large part of the mainstream media said "she even proved a theorem! and it was published!" but nobody explained what the theorem was, because it takes a few pages of exposition. But Tao does a good job with it, getting in some nice statistical mechanics on the way. (The paper itself, "“Percolation and Gibbs states multiplicity for ferromagnetic Ashkin-Teller models on Z2” " is available at McKellar's web site; I suspect this is the only instance of a mathematical theorem on a celebrity's web site, at least for suitable definitions of "celebrity". The theorem, by the way, says that the Curie temperature and the critical temperature for site percolation are equal in the Ashkin-Teller model on the square lattice.
Scott Morrison at Secret Blogging Seminar (which is becoming less secret every day) gives us grepping the Knot Atlas, which is about the Knot Atlas. The Knot Atlas is basically Wikipedia for knots. (The Knot Atlas, by the way, has no entry called "knot" or something similar that I could find. It doesn't define knots, it only tabulates them.) A knot is not exactly what the layperson would think it is; a mathematical knot is an embedding of a circle in 3-space, so the string has no ends. (Of course, one can represent any knot tied with an actual rope as a mathematical knot by splicing the ends together.) This is a continuation of the old Knot Atlas; it's a lot easier for anyone to edit a wiki. It is simultaneously a database of knot invariants, and the post I linked to is about how to extract that data.
One of the things that has always amused me about knot theory is Lord Kelvin's theory that atoms were knots; but psuedoscience often begets science. I once worked at the Department of Alchemy.
Another such wiki is qwiki for quantum physics. I don't know of any more but they seem like they could be useful, as references to various mathematical areas built by the community that actually uses them. Wikipedia plays this role to some extent, but its main function is to be a general-purpose encyclopedia so the technical details often don't make it in.
I find myself wondering if people have used wikis (with login, of course) to write mathematical papers with collaborators. (Or even alone! Since most wiki software stores old versions of pages, you could easily throw out arguments that you didn't like, confident that if they contained some seed of truth they could be recovered later.)
At Casting Out Nines, via Vlorbik: inside facts about calculators. The big one is that nobody in the "real world" actually needs a calculator; students generally have computers now, and the student versions of Maple and Mathematica are similarly priced to graphing calculators; therefore if it makes sense for students to use any sort of calculator, it should be what's usually called a "scientific" calculator. I agree. However, I have a ten-year-old TI-83 which I know pretty well, and I generally keep it around -- for when I'll be working in a coffee shop and don't feel like taking my laptop with me, or for when my officemate is using the computer. But unquestioning faith in the calculator is a bad thing. You can also read Eric Schechter's The Most Common Errors in Undergraduate Mathematics. Would it be possible to compile an analogous list for graduate students, or research mathematicians? Or do those of us at these levels make too many different errors for this to be possible?
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