Rick Miranda has written the clearest introduction to the algebraic geometry’s linear systems I’ve ever seen, in his article Linear Systems of Plane Curves from the February 1999 Notices of the AMS. He’s also written a much more technical article on algebraic surfaces which provides a good summary of the different classes of the Enriques classification.

I spotted this paper, Multiple polylogarithms, polygons, trees and algebraic cycles, on ArXiv. It relates the values of certain iterated integrals to incredibly complex and abstract objects in algebraic geometry.

Chris Peters has written two introductions to the classification of complex algebraic surfaces: a long version and a short one. The long version introduces the necessary background in complex manifolds and sheaf cohomology, while the short one skips right to the Enriques classification.

While I was looking for information on the Lewy equation, I found some introductory material on PDEs:

• Lectures notes by T. W. Körner
• Lecture notes by Joshi and Wassermann
• The course page for Methods of Numerical Simulation in Fluids and Plasmas, which includes a detailed notes on PDEs.
• The course page for Computational Methods, which includes lecture notes on PDEs and other topics.
• An essay on PDE as a unified subject

The Lewy equation is an example of an inhomogenous linear partial differential equation that has no solutions. Note that we’re not imposing any boundary-value or initial-value conditions on the equation; the equation simply has no solutions. The proof that it has no solutions is a surprisingly simple application of complex analysis. (Also available in postscript.)

The paper Fifty years of local solvability surveys the development of the theory (known as local solvability) in the wake of Lewy’s discovery. Numerical linear algebra and solvability of partial differential equations describes an analogy between local solvability and numerically computing matrix eigenvalues.

Lieven Le Bruyn has posted his vacation reading. I was planning on eventually writing something about Victor Ginzburg’s Lectures on Noncommutative Geometry, which is a survey of noncommutative geometry. Berest and Chalykh’s A∞ modules and Calogero-Moser Spaces. A∞ algebras are a generalization of algebras where the multiplication goes horribly wrong. Calogero-Moser space is a space that parametrizes right ideals in the Weyl algebra. These are two topics that I’d like to learn, so it looks interesting.

Topology Atlas is a portal site for topologists. It’s most intriguing feature is Topology Q+A, a collection of discussion forums for mathematics questions. The possibilities include Ask a Topologist and Ask an Analyst. For example, here’s Abhijit Dasgupta’s answer to a question about non-Borel measurable sets.

I was hunting for lecture notes on (measure-theoretic) probability, and I found a couple of nice links:

• Rich Bass has posted succinct introduction to probability as well as lecture notes on other topics in probability.
• Ivan Wilde has a series of lecture notes in real analysis, including one set on Measure, Integration, and Probability. He also covers C* algebras and von Neumann algebras, among other topics.