I was poking around on Wikipedia, when I came across the page for Runge’s phenomenon. Runge found an example of a smooth function such that if you interpolate it by a high-degree polynomial at fixed points over a finite interval, the approximation of the function by the polynomial becomes very bad. In fact, in the limit as the degree of the interpolated polynomial goes to infinity, the maximum difference between the function and interpolated polynomial also goes to infinity. Interpolating with polynomials is harder than it looks.

Peter spots two preprints that give two different proofs of a major open problem in algebraic geometry: the finite generation of the pluricanonical ring:

• Existence of minimal models for varieties of log general type
• A General Non-Vanishing Theorem and an Analytic Proof of the Finite Generation of the Canonical Ring

By a standard construction in algebraic geometry, this would give a particular embedding of a variety as a subvariety of projective space. Apparently for varieties of general type, this would give something even better: minimal model of the variety.

With all my fancy book-learning and abstract axiomatic thinking, there are a lot of classical results that I’ve never learned. I just ran into an interesting classical theorem by Sylvester about symmetric matrices. Let A_n be the submatrix of the first n rows and columns. When you complete the square on the associated quadratic form, the coeffients of each square term is of the form

(det A_n)/(det A_n-1).

From this it follows that the matrix is positive definite if and only if each (det A_n) is positive.

The criterion is not practical for large matrices, but it does imply one interesting theoretical result: the set of positive definite matrices is a real semialgebraic set.

Warning: if your relatives ever find out about this website, you’ve sentenced yourself to a lifetime of receives its wares for birthdays and holidays.