Completed EU Project

SOMPATY

Spectral Optimization: From Mathematics to Physics and Advanced TechnologY

About the Project

SOMPATY was an international research collaboration bringing together universities and research institutions from Azerbaijan, the Czech Republic, Germany, Kazakhstan, Ukraine and Uzbekistan.

Coordinator: The tasks of the Coordinator include: monitoring the project, acting as the official, contact with the European Commission, chairing the Supervisory Board.

Supervisory Board: The Supervisory Board is composed of leading staff representatives from all eight institutions. The Board is responsible for the coordination of the deliverables. This includes assessment of progress reports, maintenance of work plans, and conflict resolution.

Coordinator: The tasks of the Coordinator include: monitoring the project, acting as the official, contact with the European Commission, chairing the Supervisory Board.

Supervisory Board: The Supervisory Board is composed of leading staff representatives from all eight institutions. The Board is responsible for the coordination of the deliverables. This includes assessment of progress reports, maintenance of work plans, and conflict resolution.

Organization structure

Partners, WP, and Topics

SOMPATY was an international research collaboration bringing together universities and research institutions from Azerbaijan, the Czech Republic, Germany, Kazakhstan, Ukraine and Uzbekistan.

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.

The project focuses on an intensive staff exchange, which will lead to collaborative research and training between universities and research organizations from Azerbaijan, the Czech Republic, Germany, Kazakhstan, Ukraine, and Uzbekistan.

The task is organized into four Work Packages with 154 person months.
Several important project events are scheduled:
- 4 scientific workshops (Tashkent’23, Almaty’24, and Kyiv’25)
- 2 summer schools
- public outreach activities

Specific objectives and the relevance of the research & innovation action
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

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.

Objectives

The collaborative research is organized within the Work Packages (WPs).

WP 1 Numerical schemes for indefinite Sturm-Liouville problems & spectral analysis of signals

Numerical schemes for the non-real spectrum of indefinite Sturm–Liouville operators

PT-symmetric operators on complex contours
Perturbation theory for differential algebraic equations
Re-design of integrated circuits

WP 2 Linear & nonlinear models of branched structures

Spectral properties of networks, in both classical and leaky models; analysis and control
Beyond self-adjoint dynamics: dissipative and nonlinear networks

WP 3 Spectral theory of singular perturbations and inverse problems

Spectral analysis of differential operators with distribution coefficients.
Solvability of ill-posed problems of normal and generalized normal type for non-local boundary value problems
Spectrum of elliptic differential operators with nonlocal and non-coercive boundary conditions.
Spectral theory of quantum-mechanical operators with nonlocal potentials.
Direct and inverse spectral problems for quantum graphs with nonlocal potentials.
Optimization of spectral data in inverse spectral problems for special matrices.
Direct and inverse scattering problems for hyperbolic equations, application to the theory of solitons.

WP 4 Approximation theory & theory of function spaces

Localization properties of bases and frames on the multivariate torus
Construction of bases and frames on more general manifolds
Algorithms and their approximation order for non-linear and adaptive processes based on general sampling strategies
Investigation of error and stability estimates for the different function spaces involved.

WP 5 Dissemination, exploitation and outreach

Dissemination activities (months 21-66): Preparation of the four workshops and two summer schools in the years 2020-2023 with all partner teams attending:

Exploitation activities (months 21-66): Presentation of SOMPATY outcomes in relevant professional international conferences. Exploitation plan will be updated on an annual basis, taking into account relevant project deliverables and the marketing policies of the participants.
Communication activities (months 21-66): Organization of public presentations, which will be properly advertised to generate the necessary audience interest; outreach activities; organization of project events with media participation; global access and dissemination of information about the project through the internet and the creation of the project website; videos.

WP 6 Ilmenau: Project management

Project Coordination (months 21-66): Permanent management of the project actions: circulation of information between the partners; organization of Supervisory Board meetings; coorganization of workshops and other communication events; development of strategic plans and tactical schemes for effective project performance; supervision of the progress of the project; overall legal, contractual, ethical, financial and administrative management of the consortium; preparation, updating and management of the Consortium Agreement between the participants.
Monitoring (months 21-66): Monitoring the research activities and the corresponding staff secondments of the project; monitoring the risk factors, e.g. delay, underperformance etc.
Quality management and administrative matters (months 21-66): Internal audit of the project activities, paying special attention to measurements that can guarantee the conformity of the project to the requirements of the European Charter for Researchers, the Code of Conduct for the Recruitment of Researchers, and Rules of Good Scientific Practice.

The task is organized into four Work Packages (WP) with 154 person months.

WP 1 Ilmenau: #Alm#Baku
(person months: 47; Ilmenau)

WP 2 Prague: #Alm#Tash
(person months: 48; Prague)

WP 3 Kyiv: #Baku#Tash
(person months: 48; Kyiv)

WP 4 Lübeck: #Alm#Baku
(person months: 49; Lübeck)

WP 5 Kyiv: Dissemination,
exploitation and outreach

WP 6 Ilmenau:
Project management

Project Consortium

Beneficiaries

Technische Universität Ilmenau, Ilmenau, Germany

will PT symmetry and applications to metamaterials,
quantum graphs in terms of extension theory,
perturbation and spectral theory
perturbation theory for DAEs and applications to the optimization of integrated circuits

Nuclear Physics Institute of the Czech Academy of Sciences Prague, Czech Republic

will quantum graph models and PT symmetric operators in combination with metamaterials,
singular Schrödinger and Dirac operators,
leaky quantum structures
dissipative and/or many-body effects in graphs and optical fibres.

Institute for Mathematics, National Academy of Sciences of Ukraine, Ukraine, Kyiv

spectral theory
inverse problems and
differential equations.

Universität zu Lübeck, Lübeck, Germany

will all data-driven approaches for the numerical solution of the operator equations,
fast algorithms for different approximation methods and stability,
convergence and localization results,
non-linear and adaptive variants for all approximation theoretical results.

Institute of Mathematics of the University of Georgia (IMUG), Tbilisi, Georgia

Partners

Institute of Mathematics and Mathematical Modelling, Almaty, Kazakhstan

non-local boundary value problems for hyperbolic integro-differential equations
nonlocal boundary value problems.
Qualitative properties of boundary value problems for parabolic integro-differential equations

Turin Polytechnic University Tashkent, Tashkent, Uzbekistan

Linear and nonlinear PDEs on metric graphs
PT-symmetry and soliton solutions
Numerical solutions of PDEs on branched domains

Activities & Highlights

Thematic Workshops and Summer Schools

Kyiv, Ukraine

May 12-16,2025

International Workshop within the framework of the HORIZON-2020 project SOMPATY

Institute of Mathematics of the National Academy of Ukraine organises one-week workshop within the framework of the EU Project SOMPATY "Spectral Optimisation: From Mathematics to Physics and Advance Technology" whose aim consists of disseminating the ideas/results between Ukrainian and foreign project teams who suffer a lack of direct `offline' communications due to military aggression of the russian federation, as well as involving new scientific partners/teams. An important goal should be providing an opportunity to young scientists from Ukraine to report their results at a high international level with a forthcoming perspective of establishing collaborative contacts. The Workshop intends to held in a hybrid form with NO registration fee for Ukrainian speakers even though they do not belong to Ukrainian team of SOMPATY. Plenary (invited) speakers would deliver 30-60min lectures but other participants are offered to deliver 15-20min talks. No special sections are scheduled. All should hear all.

→ Plenary lectures & Talks

→ Work Program

Almaty, Kazakhstan

September 11-18, 2024

International Summer School (Sept 11-14) and Conference (Sept 16-18)

on
→ Evolution Equations, Approximation and Spectral Optimization

There are some photos from the SOMPATY summer school and conference in Almaty available on the website: https://sites.google.com/view/eeaso-2024/photo-gallery?authuser=0

Samarkand, Uzbekistan

September 11-20, 2023

International Summer School and Conference

→ Mathematics for the Micro/Nano-World: From soliton dynamics, nonlinear optics to quantum science and technology

Thematic Seminars

13th Sompaty seminar

Thursday, February 17 at 9 am

Illia Karabash

TU Dortmund and IAMM of NAS of Ukraine

Title of the talk:
A description of m-dissipative boundary conditions for Maxwell operators and its electrodynamics applications

Abstract

Motivated by modeling of leaky optical cavities, we give a description of

all m-dissipative boundary conditions for Maxwell operators in Lipschitz domains.

If time allow us, it is planned to show applications of this description to Leontovich boundary conditions

and to the cases of degenerate and random impedance coefficients.

The talk is based on a joint research with Matthias Eller.

12th Sompaty seminar

Monday, January 24 at 11:30 am

Martina Hentschel

Institute for Physics, TU Chemnitz, Germany

Title of the talk:
PT symmetry in mesoscopic systems

Short discussion

Monday, January 10 at 11:30 am

Short meeting

in order to coordinate our activities.

11th Sompaty seminar

Thursday, November 25 at 9am

Alexandra Antoniuk

Institute of Mathematics of the National Academy of Sciences of Ukraine

Title of the talk:
🗎 Pseudo-differential equation with spatial p-adic variables and some of their applications

10th Sompaty seminar

Thursday, November 11 at 9am

Elvin I. Azizbayov

The Academy of Public Administration under the President of the Republic of Azerbaijan, Baku

Title of the talk:
🗎 Nonlinear Inverse Problem for the Equation of Propagation of Longitudinal Waves with non-self-adjoint Boundary Conditions

9th Sompaty seminar

Thursday, October 21 at 9am

Carsten Trunk

Institut für Mathematik, Technische Universität Ilmenau

Title of the talk:
🗎 Sturm-Liouville, PT-symmetric operators and differential algebraic

8th Sompaty seminar

Thursday, September 30 at 9am

Jambul Yusupov

Laboratory for Advanced Studies, Turin Polytechnic University in Tashkent

Title of the talk:
Modeling and design of PT-symmetric networks: Quantum transport, soliton dynamics and dissipation effects

Abstract

We discuss extension of PT-symmetric quantum physics and soliton theory to the case of networks. In particular, we provide an approach for designing
PT-symmetric quantum graphs. Also, we discuss the problem of sliton
dynamics in networks. Relation of the both problems to the issue of
dissipative networks will be also discussed.

7th Sompaty seminar

Thursday, September 9 at 9 am

Kateryna Pozharska

Institute of Mathematics of NAS of Ukraine, Kyiv, Ukraine

Title of the talk:
Sampling recovery of multivariate functions

Abstract

We will discuss the recovery of multivariate functions from reproducing kernel Hilbert spaces in the uniform norm. We study a certain weighted least squares algorithm that uses the so-called ”standard information”, i.e., finite number of function values at some points. The main advantage of our approach is that the nodes are taken at random w.r.t. a tailored distribution and simultaneously for the whole class of functions. We show that the considered deterministic algorithm leads to near optimal results in several relevant situations. We additionally apply a recently introduced sub-sampling technique to further reduce the sampling budget.

6th Sompaty seminar

Thursday, August 19 at 9am German time

Anar T. Assanova

Institute of Mathematics and Mathematical Modeling, Almaty, Kazakhstan

Title of the talk:
Constructive methods for solving nonlocal boundary value problems for integro-differential equations PT-symmetry

Abstract

This talk is devoted to boundary value problems for Fredholm integro-differential equations and constructive methods for solving them. We will discuss the effectiveness of the Dzhumabaev parametrization method for solving nonlocal boundary value problems. We consider the questions on existence and uniqueness of solutions to boundary value problems for hyperbolic Fredholm integro-differential equations. We also present our recent research results on boundary value problems for Fredholm integro-differential equations and numerical and approximate methods for solving boundary value problems with parameter. We report about some results of boundary value problems for parabolic Fredholm integro-differential equations.

5th Sompaty seminar

Thursday, August 5 at 9am German time

Jürgen Prestin

Universität Lübeck

Title of the talk:
Multivariate Trigonometric Wavelets and Shearlets

4th Sompaty seminar

Thursday, July 22 at 9am German time

Fikret Aliev

Institute of Applied Mathematics, Baku State University, Baku, Azerbaijan

Title of the talk:
🗎 Some mathematical problems and their solution for problem of oscillatory systems with liquid dampers

3rd Sompaty seminar

Thursday, July 1 at 9 am

Pavel Exner

Doppler Institute for Mathematical Physicsand Applied Mathematics, Prague

Title of the talk:
Quantum graphs: time-reversal non-invarianceand PT-symmetry

Abstract

The talk concerns effects coming from a violation of time-reversal symmetryin the vertex coupling of quantum graphs, focusing on the situation when theasymmetry is maximal at a fixed energy. It is shown that such a coupling hasa topological property: scattering at the vertex in the high-energy regimedepends substantially on the vertex parity. We explore implications of thisfact for spectral and transport properties in several classes of graphs, bothfinite and infinite periodic ones. We also identify a class of vertex couplingexhibiting a nontrivialPT-symmetry despite being self-adjoint. The resultscome from a common work with Marzieh Baradaran, Jiří Lipovský, andMiloš Tater.

2nd Sompaty seminar

Thursday, June 10 at 9am German time

Oleksandr Burylko

Leading researcher, Dep. of Differential equations and oscillation theory of the Institute of Mathematics NAS of Ukraine

Title of the talk:
🗎 Collective dynamics in networks of phase oscillators

Abstract

The talk is dedicated to the description of collective dynamics in the networks of phase oscillators of the Kuramoto type. I plan to show how the architecture and symmetry of the network, individual dynamics, and the interaction functions affect the emergence, structure, stability, and bifurcations of different collective regimes (synchronization, clustering, splay states, antiphase regimes, slow switching, chimera state, etc.).

1st Sompaty seminar

Thursday, May 20 at 9am German time

Davron Matrusulov

Laboratory for Advanced Studies, Turin Polytechnic University in Tashkent

Title of the talk:
🗎 Tunable quantum networks: Modeling and design transparent and PT-symmetric quantum graphs

Abstract

In this talk I will briefly introduce the concept of quantum graph as model of branched quantum structures and discuss how to make their branching points (vertices) transparent, i.e. providing reflectionless transmission of waves/particles through these points. Also, I willl introduce concept of PT-symmetric (non-Hermitian) quantum graphs and provide sketch for their experiemntal realization.