I will talk about some recent progress on the problem of constructing 3D Euclidean Yang-Mills theories. This is based on joint work with Sky Cao.

I will talk about the micro-canonical invariant measure for the discrete nonlinear Schrödinger equation on a torus in the mass-subcritical regime, and prove that a random function drawn from this measure is close to the ground state soliton with high probability. This proves that “almost all” ergodic components of this flow have the property of convergence to a soliton in the long run, which is a statistical variant of what is sometimes called the soliton resolution conjecture