Pierre Perruchaud

As an illustrative first example, consider the zero set of a smooth function on the sphere. If the function is close to the constant 1, then the zero locus is empty; likewise if it is close to the constant -1. However, by the intermediate value theorem, if we deform the first function into the second, then at some point the zero set is going to be non-empty. Under reasonable assumptions, it will first be a point, then a loop, then it will grow in size, then possibly the topology will become richer (more connected components), then it will start to shrink, until it is just a loop, shrinking to a point, and disappearing.
We will introduce this model and similar one, such as the rubber band moving on a table, which can go over itself but never break. The fundamental questions we will ask are what kind of topological phases can these objects present, and how and when do they transition between those.