This talk is motivated by interest in Crouzeix’s conjecture for compressions of the shift with finite Blaschke products as symbols. Specifically, in this setting, Crouzeix’s conjecture suggests a related, weaker conjecture about the behavior of level sets of finite Blaschke products. I’ll discuss this level set conjecture in several cases, though the main case of interest will involve uncritical finite Blaschke products. Here, the geometry of the numerical ranges of their associated compressions of the shift has allowed us to establish the conjecture in low degree situations (n=3, n=4, n =5 with a caveat). Time permitting, I’ll explain how these geometric results also give insights into Crouzeix’s conjecture for the associated compressed shifts. This talk is based on joint work with Pam Gorkin.

The video of this talk is available on the IISc Math Department channel.