There exists something called the Harvard College Mathematics Review, which "is a semiannual journal of expository mathematical articles written and edited by undergraduates." I learned about this from Vishal Lama's blog. (Conveniently, the how to link to a blog question is almost irrelevant here.)
The faculty feature article by Noam Elkies which was in their first issue, about the abc conjecture, is quite interesting. (I especially like the heuristics given therein for the number of rational points lying on certain curves and other solutions to diophantine equations -- concerning such questions as Mordell's conjecture and Fermat's last theorem -- which shouldn't surprise longtime readers.)
There have been two issues so far: the first issue and the second issue.
You know it's got to be good -- or at least it looks good from the table of contents and the few links I clicked on -- because I went to MIT for undergrad and I'm now a grad student at Penn and I'm still telling you about it. And you've got to love any publication which includes an article entitled Dunking Donuts: Culinary Calculations of the Euler Characteristic. I mean, who hasn't gotten hungry while doing topology? All those tori look like donuts.
And the editor-in-chief of the HCMR is Scott Kominers, who I've previously mentioned as one of the authors of a paper about whether or not you should wait for the bus.
Showing posts with label algebraic topology. Show all posts
Showing posts with label algebraic topology. Show all posts
02 March 2008
19 November 2007
Abusive dimensions and cohomology of Grassmannians
Yesterday I complained about the abuse of the word "dimension" by a lot of popular authors, namely the fact that they tried to order the dimensions.
Being a discrete mathematician, I don't think about dimensions all that often. But I'm taking a course in algebraic topology (it's required for my program); I have a habit of trying to twist the problems there into combinatorics problems whenever possible. Recently we were asked to compute the cohomology of the Grassmannian manifold GnRk. This Grassmannian is just the manifold of n-dimensional subspaces of k-dimensional space passing through the origin. It is of dimension k(n-k), as we can see by "dimension-counting". To obtain such a subspace we pick n orthogonal unit vectors in k-space. The first one is chosen from Sk-1 and thus we have k-1 "degrees of freedom", contributing k-1 to the dimension. The second one is chosen from a (k-2)-sphere in the (k-1)-dimensional orthogonal complement of this first vector; this contributes k-2 dimensions. And so on, down to k-n dimensions for the nth vector. But we've overcounted by 1 + 2 + ... + n-1, since we've actually chosen an n-dimensional flag in k dimensions; thus we need to quotient out by the orthogonal group SO(n), which has this dimension. Thus the dimension of the Grassmanian is
[(k-1) + (k-2) + ... + (k-n)] - [0 + 1 + ... + (n-1)] = n(k-n).
As you may have noticed, I used "dimension" in two senses there; the rigorous sense of the topological invariant which is always an integer, and the non-rigorous sense of "degree of freedom". At this point my complaint from yesterday is still valid, because I never ordered the dimensions, and that was the cardinal sin I attributed to people who don't know what they're talking about.
But then I caught myself saying things like "the cohomology of G2R5 in the second dimension is Z2." That's no good.
As it turns out, I rediscovered the fact that the dimension of Hm(GnRk) is the number of integer partitions of m into at most k-n parts, none of which exceed n; in the case I just mentiond we have m = 2, n = 2, k = 5, so we're looking for integer partitions of 2 into 2 parts which don't exceed 3. There are two of them, namely 2 and 1 + 1. This isn't too interesting, but say we want to know, for example, the dimension of H9(G3R10)? Then we can just write down all the partitions of 9 into three parts which are at most 7;
7+2+0, 7+1+1, 6+3+0, 6+2+1, 5+4+0, 5+3+1, 5+2+2, 4+4+1, 4+3+2, 3+3+3
and there are ten of them; thus H9(G3R10) = Z10. You can write down generating functions for these sequences of integers; I suppose one could even ask about the asymptotic behavior of these dimensions, although I don't think a topologist would care. (I can't find a source for this fact, but I'm actually pretty sure I've heard it before; the two obvious sources to me were Hatcher's Algebraic Topology and Stanley's Enumerative Combinatorics, which are both books I know reasonably well, and I found the place in each where this should be mentioned. Stanley just says that there is a connection to the cohomology of the Grassmannian and leaves it at that; Hatcher talks about symmetric polynomials but never sinks to finding actual numbers.)
But as I said, I caught myself saying things like "what's the second-dimensional cohomology of G2R5"? Second-dimensional. And I've heard this usage from Real Algebraic Topologists, too. So apparently what was bothering me yesterday was something more subtle than this abuse of language.
Being a discrete mathematician, I don't think about dimensions all that often. But I'm taking a course in algebraic topology (it's required for my program); I have a habit of trying to twist the problems there into combinatorics problems whenever possible. Recently we were asked to compute the cohomology of the Grassmannian manifold GnRk. This Grassmannian is just the manifold of n-dimensional subspaces of k-dimensional space passing through the origin. It is of dimension k(n-k), as we can see by "dimension-counting". To obtain such a subspace we pick n orthogonal unit vectors in k-space. The first one is chosen from Sk-1 and thus we have k-1 "degrees of freedom", contributing k-1 to the dimension. The second one is chosen from a (k-2)-sphere in the (k-1)-dimensional orthogonal complement of this first vector; this contributes k-2 dimensions. And so on, down to k-n dimensions for the nth vector. But we've overcounted by 1 + 2 + ... + n-1, since we've actually chosen an n-dimensional flag in k dimensions; thus we need to quotient out by the orthogonal group SO(n), which has this dimension. Thus the dimension of the Grassmanian is
[(k-1) + (k-2) + ... + (k-n)] - [0 + 1 + ... + (n-1)] = n(k-n).
As you may have noticed, I used "dimension" in two senses there; the rigorous sense of the topological invariant which is always an integer, and the non-rigorous sense of "degree of freedom". At this point my complaint from yesterday is still valid, because I never ordered the dimensions, and that was the cardinal sin I attributed to people who don't know what they're talking about.
But then I caught myself saying things like "the cohomology of G2R5 in the second dimension is Z2." That's no good.
As it turns out, I rediscovered the fact that the dimension of Hm(GnRk) is the number of integer partitions of m into at most k-n parts, none of which exceed n; in the case I just mentiond we have m = 2, n = 2, k = 5, so we're looking for integer partitions of 2 into 2 parts which don't exceed 3. There are two of them, namely 2 and 1 + 1. This isn't too interesting, but say we want to know, for example, the dimension of H9(G3R10)? Then we can just write down all the partitions of 9 into three parts which are at most 7;
7+2+0, 7+1+1, 6+3+0, 6+2+1, 5+4+0, 5+3+1, 5+2+2, 4+4+1, 4+3+2, 3+3+3
and there are ten of them; thus H9(G3R10) = Z10. You can write down generating functions for these sequences of integers; I suppose one could even ask about the asymptotic behavior of these dimensions, although I don't think a topologist would care. (I can't find a source for this fact, but I'm actually pretty sure I've heard it before; the two obvious sources to me were Hatcher's Algebraic Topology and Stanley's Enumerative Combinatorics, which are both books I know reasonably well, and I found the place in each where this should be mentioned. Stanley just says that there is a connection to the cohomology of the Grassmannian and leaves it at that; Hatcher talks about symmetric polynomials but never sinks to finding actual numbers.)
But as I said, I caught myself saying things like "what's the second-dimensional cohomology of G2R5"? Second-dimensional. And I've heard this usage from Real Algebraic Topologists, too. So apparently what was bothering me yesterday was something more subtle than this abuse of language.
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