While I was looking for information on the Lewy equation, I found some introductory material on PDEs:
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.
AMS Summer Institute in Algebraic Geometry is underway in Seattle. It’s a mammoth three-week conference on algebraic geometry. The first week is dedicated to the unlikely connections that have emerged between algebraic geometry and string theory.
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.
Lee Smolin has posted a philosophical article on alternative approaches to quantum gravity: The case for background independence.
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:
At the new group physics weblog Cosmic Variance, Sean Carroll has posted a defense of string theory, Two Cheers for String Theory. Chad Orzel and Peter Woit have also weighed in.
It’s basically impossible to know all of the important concepts and results in mathematics. It’s impossible to even have heard of all of the important concepts and results in mathematics. For example, I’d never heard of expander graphs, which apparently have widespread applications in combinatorics and computer science, and even have an interpretation in terms of group representations.
Michael Nielsen has a series of posts on expander graphs beginning here. For more background, he links to lecture notes on the subject by Linial and Wigderson.
Igor Dolgachev, a mathematician at the University of Michigan, has made available lecture notes on topics in algebraic geometry and physics. The lecture notes in algebraic geometry include invariant theory and what he calls “classical algebraic geometry”. He also provides an introduction to theoretical physics for mathematicians, and as well as one on string theory.