#michael freedman

“the Yang-Mills equations are nonlinear, therefore there is little hope of finding a closed-form solution.” Such a statement seems plausible. Linear differential equations with constant coefficients are the only differential equations for which a general solution is given in closed form. As often occurs in life, however, the exceptions to the rule are sometimes more interesting than the rules themselves. Let us digress from quantum physics to the motion of water, where British shipbuilder John Scott Russell noticed a solitary wave in a canal in August 1834. Neither Airy nor Stokes accepted this observation, yet in 1895 Korteweg and de Vries found an equation for a wave travelling in shallow water in one direction: u̇ + 6•u•uₓ + uₓₓₓ = 0. The KdV equation is easily solved by restricting from two independent space-time dimensions (x,t) to a single dimension x−λt — a frame matching the speed λ of a travelling wave.

In dimensions ≥5, topology and algebra match up simply. In dimensions 3 and 4, manifolds are more interesting. We used to wonder if there was some key idea from high-dimensional manifolds that would extend at least down to dimension 4, making 4-manifolds easy to understand. There is such a principle: Imbedding 2-handles in smooth 4-dimensional manifolds-with-boundary subject to a homotopy criterion.  This high-dimensional idea does work in 4 dimensions and give us insight into 4-manifolds.

Moore planning Plate Attack