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.

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 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.