Crouzeix’s Conjecture for Nilpotent Matrices
Crouzeix’s conjecture is a central open problem in matrix analysis, bounding the norm of a matrix function by twice its supremum on the numerical range. This project studied the level set Crouzeix conjecture — a weak version that applies to finite compressions of the shift.
Key contributions:
- Established the level set Crouzeix conjecture for several classes of 3×3, 4×4, and 5×5 nilpotent matrices via a geometric analysis of their numerical ranges
- Developed a Mathematica-based tool for fast visualization of numerical ranges
- Presented at the Joint Mathematics Meetings 2023 (Boston); published in Concrete Operators (2024)
Principal Investigator: Dr. Kelly Bickel