NZMRI 2026

Title: Braid groups, configuration spaces and holomorphic rigidity
 

Abstract: 

 

Many remarkable constructions in algebraic geometry can be described as holomorphic maps between various moduli or parameter spaces. For many pairs of such spaces, the only known non-trivial holomorphic maps between them come from a fairly small list of constructions. A natural rigidity question to ask is: are these the only examples of holomorphic maps between the spaces? This often turns out to be related to understanding the homomorphisms between the associated fundamental groups.

 

We will focus on a particular special case: configuration spaces of points in the complex plane. An example of a remarkable construction in this context is given by Ferrari's solution to the quartic equation: we can view this as holomorphically producing a set of three distinct complex numbers from a set of four distinct complex numbers. I will describe the topology of these configuration spaces, including their fundamental groups: the braid groups, which are examples of mapping class groups. Later I will discuss what we know about the homomorphisms that can occur between braid groups, and some beautiful results in complex analysis that allow us to leverage this group theory to answer holomorphic rigidity questions. This will lead us to a result of Schillewaert and I that, in particular, gives a nice characterization of the aforementioned map due to Ferrari. Time permitting I will also survey some other results in the world of holomorphic rigidity.