Ars Mathematica

People of Earth,

Now that this has happened, time is short. I can only assume that the inhabitants of the planet around Gliese 581 have left us unmolested up to this point is that as long as we did not know of their existence we were no threat. Now that the word is out, I can only assume that their long-prepared invasion fleet is under way. Fortunately, we have 20.5 years until word reaches them, and another 20.5 before their planet-killing machines can acheive Earth orbit, so we must use the 41 years of peace left to us to prepare. I am not a crank.

Sincerely,

Walt

In this thread on his weblog, Terry Tao mentioned an exciting paper by Noga Alon. The paper explains a result that Alon calls the Combinatorial Nullstellensatz. The ordinary Nullstellensatz relates algebra and geometry over algebraically closed fields. Consider the set of common zeroes of a system of (multivariate) polynomial equations over the field. Then a polynomial vanishes on that set if and only if a power of the polynomial is in the ideal generated by the system of equations.

It is easy to see that no such simple result holds over non-algebraically closed fields. Alon is able to prove an analogue of the Nullstellensatz in a very special case, so special that it is not of particular interest of itself. But he is able to use it to give new short proofs of many existing results, such as the Chevalley-Warning and Cauchy-Davenport theorems, as well as many results in combinatorics.

I’ve come across this guide to quantum field theory textbooks. Since QFT is on my to-do list of things to learn before I die, this list may come in handy.

People always post interesting links in the comments to Scott Aaronson’s weblog. For example, the other day Paul Beame posted two links that explain the connections between random walks on graphs and electrical networks. One is a complete book on the subject by Doyle and Snell. The other is an article by Chandra, Raghavan, Ruzzo, Smolensky, and Tiwari that further develops the theory.

In a comment thread at n-category cafe, John Baez has linked to the electronic version of a big bowl of ice cream. Investigating the link in his comment, I came across the web page of Francis Sergeraert, who has linked to his papers and talks.

Sergeraert and his collaborators have pioneered a program of computational algebraic topology, and it is amazing what they have already acheived. For example, they have developed effective versions of the Serre and Eilenberg-Moore spectral sequences.

These kinds of algorithms exert a powerful hold on my imagination. When I first tried to learn commutative algebra, I found much of the subject impenetrable. Then later when I learned about Gröbner bases, I suddenly found everything I found hard to understand became easy to understand. Now the Gröbner basis algorithm is too slow to implement by hand other than toy examples, but having effectively computable toy examples was enough for me. Commutative algebra textbooks are full of toy examples, but my suspicious unconscious mind was sure that they were tricking me, and that the toy examples were not to be trusted. Learning how to compute new examples allowed me to shut my unconscious up.

The 2007 Abel Prize has been announced. The winner is S. R. Srinivasa Varadhan for his work on large deviations in probability. Large deviations are asymptotic estimates of rare events. They are of practical importance, for example, in justifying the results of statistical mechanics. Cosma Shalizi’s notebook on large deviations provides an overview and many more links.

I find it interesting that the Abel Prize has taken a turn towards the applied in recent years. The first two awards, to Serre and to Atiyah and Singer, track the expectations of pure mathematicians. In the last three years, though, one prize has gone to Peter Lax, who works in applied PDEs, and now this year Varadhan. (The other winner is Lennart Carleson.)

The April Notices of the AMS features a retrospective on Serge Lang’s mathematical career. This month’s What is…, What is… a tropical curve. A tropical curve is a curve defined over the tropical semiring, which is the real numbers with addition defined as max or min, and multiplication defined as real addition. Mathematicians are currently working on finding the analogues of classical results on algebraic curves to the tropical setting.

The fifth Carnival of Mathematics is up.

The functional analysis thread has turned into a big love-fest, but a dissenting voice has appeared from an unexpected quarter. David Corfield spotted this interview with Gian-Carlo Rota and David Sharp. Rota says:

Combinatorics is an honest subject. No adèles, no sigma-algebras. You count balls in a box, and you either have the right number or you haven’t. You get the feeling that the result you have discovered is forever, because it’s concrete. Other branches of mathematics are not so clear-cut. Functional analysis of infinite-dimensional spaces is never fully convincing; you don’t get a feeling of having done an honest day’s work. Don’t get the wrong idea-combinatorics is not just putting balls into boxes. Counting finite sets can be a highbrow undertaking, with sophisticated techniques.

This raises a disturbing possibility: I like functional analysis because it provides yet another way to avoid an honest day’s work.

Functional analysis took the fun out of analysis. Discuss.