Room 1.12 Building VII
   
Abstract: In this talk we will address the lifting problem for the Černý conjecture: namely, whether the validity of the conjecture for a quotient automaton can always be transferred (or “lifted”) to the original automaton. Although a complete solution remains open, we show that it is sufficient to verify the Černý conjecture for three specific subclasses of reset automata: radical, simple, and quasi-simple.  Our approach relies on establishing a Galois connection between the lattices of congruences and ideals of the transition monoid. This connection not only serves as the main tool in our proofs but also provides a systematic method for computing the radical ideal and for deriving structural insights about these classes.  
This is a joint work with Prof. Emanuele Rodaro.

Wave scattering is present in many applications regarding non destructive testing, namely medical imaging. In particular, in the context of inverse scattering problems, the goal is to find unknown properties of the obstacle (scatterer) from the knowledge of the incident field and the measurement of the scattered field or far-field pattern. In this talk, we will focus on numerical methods to solve mathematical models for time-harmonic scattering problems. We will focus on new approaches to ensure stability of the method of fundamental solutions to solve the direct problem and in numerical methods that deal with the ill-conditioning and nonlinearity of the inverse problem, in a time-harmonic wave propagation context. Finally, we will consider a numerical approach for a toy model for elastography, a medical imaging modality used for biological tissue imaging, focusing on their applications in the retina.