The smooth mean curvature flow often develops singularities, making weak solutions essential for extending the flow beyond singular times. Among various weak formulations, the level set flow method is notable for ensuring long-time existence and uniqueness. However, this comes at the cost of potential fattening, which reflects genuine non-uniqueness of the evolution after singular times. With Alec Payne, we establish an intersection principle comparing two intersecting flows and prove that level set flows satisfy this principle in the absence of non-uniqueness.

I will present a new method to engineer integrable models in 4d with higher genus spectral parameters. The method has a twistorial origin - by working on a branched covering of twistor space, I show how one can derive deformations of holomorphic BF theory on twistor space which descend to elliptic and hyperelliptic models on R^4 via the Penrose transform. I show how one can bootstrap the Penrose transformed actions using symmetry and integrability to find deformations of self-dual Yang-Mills theory. I will also discuss some novel deformations of a BF type description of Hitchin’s equations. This is based on my recent paper: 2509.12486