Abstract: A cylindrical surface of soap film suspended between two circles will eventually snap if the circles are moved far enough apart. Similarly, if the data defining a complex algebraic manifold is adjusted in certain ways the manifold will break. Particularly severe cases of this are called large complex structure limits and these have proved to be very important both in theoretical physics (string theory, super symmetry) and algebraic geometry. A key insight of Yan Soibelman and Fields medalist Maxim Kontsevich is that large complex structure limits can be understood as the manifold collapsing onto the boundary of a polytope or, more generally, a simplicial complex. After giving a non-technical account covering some of this background I will explain that in order to get a finer understanding of it one is led to a partial differential equation on the boundary of a polytope and present the first general existence and uniqueness results for this equation (joint work with Mattias Jonsson, Enrica Mazzon and Nick McCleerey).
