Harvard-MIT Algebraic Geometry Seminar: Tuples of matrices (after Charlie Wu)

Consider the space of n-tuples of matrices whose product is the identity, up to simultaneous conjugation, and where the matrices are constrained to lie in some collection of fixed conjugacy classes. This is an example of a “relative character variety,” in this case of the projective line minus n points. I’ll discuss work of Charlie Wu (in his PhD thesis) describing some of the topology of these spaces, as well as relative character varieties of higher genus curves, and in particular a classification of the compact components of the real points of these spaces. I’ll also discuss joint work in progress that further studies the rich geometry of these compact components and their mapping class group dynamics.