Blaschke Products, Level Sets, and Crouzeix's Conjecture

Dr Kelly Bickel (Bucknell University) will present her research in Analysis.

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 several low degree situations.

Time permitting, I’ll explain how these geometric results also give insights into the Crouzeix’s conjecture itself for the associated compressed shifts. This talk is based on joint work with Pamela Gorkin.

The talks are held using Zoom. Anyone interested in attending (outside the School of Mathematics) should email Ben Sharp at to request zoom coordinates.