Crouzeix's conjecture is among the most intriguing developments in matrix theory in recent years. Let $\|\cdot\|$ denote the 2-norm and let $W(A) = \{ (Au,u) : \|u\| = 1\}$ denote the field of values (numerical range) of the $n\times n$ matrix $A$. Crouzeix's 2004 conjecture postulates that for any function $f$ analytic on $W(A)$, $\| f(A)\| \le 2 \sup_{z\in W(A)} |f(z)|$, i.e., the spectral norm of a function of a matrix $A$ cannot be more than twice the magnitude of that function on the field of values of $A$. This bound has significant implications for applications ranging from iterative methods for the solution of linear algebraic systems (where $f$ is a polynomial of degree $k$ at the $k$th iteration) to the analysis of continuous time linear dynamical systems (where $f(z) = e^{tz}$). Remarkably, Crouzeix proved in 2007 that the inequality above holds if 2 is replaced by 11.08. Furthermore, it is known that the conjecture holds in a number of special cases, including $n=2$. This minisymposium will review what is already known, present related new results, and offer perspectives on possible approaches to resolving the conjecture in the future.

### Monday 11/07, 10:30–12:30

- Mon 10:30–11:00, SW RaadzaalAbout my conjecture, Michel Crouzeix (Université de Rennes 1).
- Mon 11:00–11:30, SW RaadzaalInvestigation of Crouzeix's Conjecture via Nonsmooth Optimization, Michael Overton (New York University).
- Mon 11:30–12:00, SW RaadzaalNear Normal Dilations of Nonnormal Matrices, Anne Greenbaum (University of Washington).
- Mon 12:00–12:30, SW RaadzaalOn the approximation of positive definite Hankel matrices., Bernhard Beckermann (Univ. Lille).