Jim Gray, the 1998 Turing Award winner, has gone missing on a sailing trip off the coast of California to scatter his mother’s ashes.
His book on the inner workings of databases, Transaction Processing: Concepts and Techniques, is very good.
Update. Cosmic Variance has more about Gray. He was a major contributor to the Sloan Digital Sky Survey.
Commenter tdstephens3 in this thread at Mathematics Under the Microscope:
Mathematicians aren’t born from school math competitions in the same way that poets do not grow out of spelling bees.
At the rate I’ve been writing up this post, I’m surprised I finished it before March. The January Notices of the AMS have been out for a while. The feature article, Homological Sensor Networks, describes an application of homology to network design. I predict computational homology will be a major growth area for applied mathematics in the future.
What is… a projective structure introduces manifolds that are locally modelled after projective space. There’s also a review of Fearless Symmetry, which is a popularization of advanced number theory (going so far as to talk about the relationship between number theory and representations of Galois groups, apparently). My library has this book, so I plan taking a look to see how the authors do.
(The February Notices are already online, but I’ll save a post for that when I finally get a chance to look at it. If the March Notices are already online, I don’t want to know about it.)
Sometimes I think I have a handle on Banach spaces. Then I contemplate the example of Tsirelson space, which is a Banach space that does not contain as a subspace any classical sequence space (c0 or lp).
This is somewhat off-topic for this site, but considering its importance, I thought it was important to post it.
The novel Snowcrash is a dystopian cyberpunk future where civil society has completely shattered and everyone lives in storage sheds. And it’s coming true. My advice? Buy shotgun shells and head for the hills.
In the novel, the Federal government (which no longer has any power, and survives as a contracting agency for large projects in need of bureaucrats) sends forms to its employees via computer that employees must read and fill out. The employees know that if they do it too quickly, they will get in trouble for not following the form closely enough, in which case they will be in violation of the law. Professors at Southern Illinois University at Carbondale (including a math professor, Marvin Zemin) are in this exact situation. And when I read the novel, this stuck in my mind because it was so implausible…
(Story spotted on Uncetain Principles.)
I was glad to see that Wikipedia’s page for the Baker-Campbell-Hausdorff formula actually explicitly states the formula. When I was first learning the subject of Lie groups and algebras, the authors would only show the first few terms, and then an ellipsis. It always left me with the impression that the actual formula was so hideous that no one ever mentioned it explicitly, but only passed over it in discreet silence.
I just ran across a quote by Hilbert from 1930:
The real reason for Comte’s failure to find an unsolvable problem is, in my opinion, that an unsolvable problem does not, altogether, exist.
I assume that Comte is Auguste Comte, the sociologist, but I don’t know what remark Hilbert is alluding to. Gödel published his incompleteness theorem in 1931.
Last year, the entire editorial board of the journal Topology resigned in protest of the high subscription fees charged by the publisher, Elsevier.
Peter Woit notes that many of the same people have now announced a new journal, the Journal of Topology. The annual subscription price is $570. (The price for Topology was $1665.)
We have readers of all backgrounds here at Ars Math, so I thought I would experiment with a more expository post than usual. Commenter Wendell Dryden is teaching himself the Chinese remainder theorem.
The result has been considerably generalized (as the Wikipedia entry makes clear), and one variant is easier to understand (I think): polynomial interpolation. Given n numbers xi on the x-axis, and n numbers, yi, on the y-axis, you can always find a polynomial p such that p(xi) = yi. The steps in solving this problem and the integer congruence problem are similar, and both problems can be solved by using the extended Euclidean algorithm.
This analogy has been taken much further in algebraic geometry. From that point of view, an integer is no longer just a number, but actually (like a polynomial) a function. The integer, in its function guise, sends prime numbers to that integer modulo that prime. So the function 67 sends 2 to 1, 3 to 1, 5 to 2, 7 to 4, 11 to 1, etcetera. (The value eventually stabilizes, in this instance at 67, which always seemed to me must be a fact of some significance, but I’ve never seen it used for anything.)
So now you know: to an algebraic geometer, integers are functions. Mathematics is like drugs, but cheaper.
Here’s an analogy that I try to complete from time to time.
Integers:Reals::Free Group on Two Generators:?
(Under addition, the integers are the free group on one generator.) I’m not precisely sure what properties of the construction of the reals I’m trying to generalize to the right-hand side of the analogy, beyond the fact that the answer should be a group that is a path-connected topological space.
Two candidates I’ve considered are: 1) certain sets of paths on the plane (which is naturally a groupoid, but you can bully it into being a group) or 2) the Lie group corresponding to the free Lie algebra on two generators (I don’t know in what sense, if any, such an object exists).