The aim of our project is inspired by the innovations that have produced new materials and devices with unusual properties. The theoretical basis of this research is quantum mechanics, naturally in combination with electromagnetism theory. The dynamics of quantum systems is governed by the Schrödinger equation, and the fundamental information is encoded in the spectrum of the corresponding Hamiltonian – or energy – operator. The system in question has a complicated structure, and thus the equations that arise require new mathematical tools to solve them. This can rarely be done explicitly. The same is true in other situations: for instance, in electromagnetic systems with a source and drain, the concept of PT symmetry is crucial, or, in chip design, information about the spectrum of a first-order matrix polynomial is extremely important and corresponds to a complicated system of differential algebraic equations. Most of these problems are connected with the solvability of linear or nonlinear equations and, in particular, of associated boundary value problems. This requires, among other things, rigorous or approximate solutions for such equations. In all these problems, there is a deep interplay between physics, pure and applied mathematics and applications from engineering. A common tie from the mathematical viewpoint is the spectrum which plays also a key role in quantum mechanics. A profound understanding of the relationship between the spectrum, geometry, and topology of the described physical or technological objects, as well as of the influence of their deterministic or random perturbations, is needed if they are to be designed in a better way than just using trial and error.
Key research questions include
WP 1: Numerical schemes for indefinite Sturm-Liouville problems & spectral analysis of signals
WP 2: Linear & nonlinear models of branched structures
WP 3: Spectral theory of singular perturbations and inverse problems
WP 4: Approximation theory & theory of function spaces
The aim of the SOMPATY project is to strengthen research ties between Europe and the Commonwealth of Independent States (CIS), particularly the CIS countries in Asia Minor and Central Asia.
Research and innovation objectives
- To model optical fibre networks, nanoscale waveguides and branched structures, nanomaterials and metamaterials including dissipative materials.
- To contribute to the understanding of novel types of metamaterial structures.
- To deepen spectral optimization methods in inverse problems for integrated circuits.
Networking objectives
- The transfer of knowledge and sharing of experience between the participants, in order to create innovations based on the know-how of the partners in the CIS countries in Asia Minor and Central Asia and the European CIS country of Belarus.
- Revealing the potential of the Hidden Champions in the CIS countries.
- International collaboration and secondments to the CIS countries in order to benefit from the knowledge and good practice of the Hidden Champions.
- The establishment and maintenance of a sustainable innovation network. The hope is that a broader European research community will benefit from a cascading effect.
