NWO Open Competition – Grant M20.292
This project is centered around —but not limited to— Crouzeix’s conjecture, a statement an optimal absolute constant in an inequality and related questions at the intersection of functional analysis, complex analysis and numerical linear algebra.
The conjecture states that the numerical range of a general square matrix with complex entries is a 2-spectral set. This property and previous attempts to resolve the conjecture relate to several topics in operator theory, such as dilation theory, functional calculus and spectral theory, as well as matrix and numerical analysis.
The aim is to study several research questions connected to this open problem, which, however, are of independent interest. This will be done by exploiting recent developments in the area, as well as established methods, such as, for instance, the interplay between complex analysis and operator theory.
Concretely this is achieved by splitting the system into three loops each with a degree of autonomy. A first one on the avatar side predicting the operator input and controlling the avatar, one on the operator side predicting the environment response and providing feedback to the operator and a third in between which connect the other two, continuously updating their models and minimizing mismatches, using the delayed connection between them.
The mathematical part of this interdisciplinary project will then focus on developing novel control techniques for use within these three loops, applying for example energy-based models such as port-Hamiltonian systems and model-free control techniques, such as funnel control.
Public outreach description:
The difference of 2 and 1 + √2 and why we should care:
Mathematicians are generally known for two things; they love puzzles and they love being picky. This project, dealing with a conjecture of French mathematician Michel Crouzeix, reflects exactly these two peculiarities. The puzzle serves to prove a certain inequality in its sharpest form, that is, with the best possible involved constant. Such inequalities are used to predict errors when solving large systems of equations. So far, the best known constant is 1 + √2 ≈ 2.4142 and the claim is that this value can be replaced by 2. Computer simulations seem to confirm that this is true indeed but only a mathematical proof will solve the puzzle.