Ars Mathematica

Penny Smith has posted a preprint to arXiv, Immortal Smooth Solution of the Three Space Dimensional Navier-Stokes System that, if correct, would solve one of the Clay Institute’s Millenium Problems. Christina Sormani has created detailed summary of Smith’s work on PDEs and Navier-Stokes.

The Navier-Stokes equation is a set of equations that describe fluid flow in Newtonian mechanics. The equations are notoriously difficult to analyze. The existence of smooth solutions for all time (the meaning of “immortal” in the paper title) has long been an open question. One now perhaps closed.

Via Peter Woit.

Update. The paper has been withdrawn. (Via John Baez in the comments.)

I’d like to point out that the 2006 Nobel Prize in Physics was awarded for this diagram:

No word on whether the original diagram included the same caption. More information available from Backreaction. Graphic via XKCD.

The latest issue of the Bulletin of the AMS is out. The feature article is Expander graphs and their applications by Hoory, Linial, and Wigderson. Expander graphs are a kind of graph that are important in computational complexity theory; we discussed them once before. Y.S. Sinai, a mathematician who works on topics quite close to physics, has an interesting article on the cultural differences called Mathematicians and physicists = cats and dogs?.