In quantum mechanics, Heisenberg's uncertainty principle states that one cannot simultaneously measure the position and momentum of a particle. Mathematically, this uncertainty relation can be expressed by stating that a function and its Fourier transform cannot both be compactly support or rapidly decaying. Over the decades, various versions of this principle were established with  so-called localization operators playing a pivotal role. These operators  found diverse applications, from sampling theory to statistics.

The first goal of this project is to investigate   uncertainty relations for operators instead of functions. In mathematics, an operator is a linear mapping between two spaces of functions or vectors. These mappings are used, for example, in mobile communications to model a communication channel. We will use methods of quantum harmonic analysis and the theory of localization operators to explain and systematically  study the localization properties of operators.

The second goal of this project is to examine how certain characteristics of an operator can be reconstructed solely based on one (or a few) measurements of the operator's output. In this sense, operator reconstruction could, for example, involve identifying the current calibration of a hearing aid or detecting an object using radar. Classical reconstruction methods aim to restore signals or operators as accurately as possible. However, this is often unnecessary, as certain parameters often already contain the essential information about the behavior of an operator. We will investigate when and how this philosophy leads to satisfactory reconstruction results. To do so, we will build on the theory of operator identification and aim to develop randomized methods. Statistical algorithms are playing an ever increasing role in applied mathematics, as they provide reliable results despite their inherent randomness, while following a simple structure.

This project was granted as a 3.5-years FWF project, that will start approximately in October 2024, hiring one PostDoc.