About

CA18232 Mathematical models for interacting dynamics on networks (mat-dyn-net)

Many physical, biological, chemical, financial or even social phenomena can be described by dynamical systems. It is quite common that the dynamics arises as a compound effect of the interaction between subsystems in which case we speak about coupled systems. This Action shall study such interactions in particular cases from three points of view:

1. the abstract approach to the theory behind these systems,
2. applications of the abstract theory to coupled structures like networks, neighboring domains divided by permeable membranes, possibly non-homogeneous simplicial complexes, etc.,
3. modeling real-life situations within this framework.

The purpose of this Action is to bring together leading groups in Europe working on a range of issues connected with modeling and analyzing mathematical models for dynamical systems on networks. It aims to develop a semigroup approach to various (non-)linear dynamical systems on networks as well as numerical methods based on modern variational methods and applying them to road traffic, biological systems, and further real-life models. The Action also explores the possibility of estimating solutions and long time behavior of these systems by collecting basic combinatorial information about underlying networks.

Areas of Expertise Relevant for the Action

• Theoretical aspects of partial differential equations
• Numerical analysis
• Operator algebras and functional analysis

Keywords

• dynamical systems on networks
• linear and nonlinear operator semigroups
• coupled systems of evolution equations
• spectrum of quantum graphs
• numerical analysis of coupled PDEs

Research Coordination Objectives

• Coordinate and direct research efforts by groups from different subdisciplines of mathematics.
• Study evolution equations on higher-dimensional multi-structures or ramified spaces, and develop the abstract theory needed for this study.
• Refine the previously known abstract techniques and develop new ones so that more general DSNs can be modeled and analyzed using mathematical tools.
• Generalize available methods to handle nonlinear problems on networks, in particular discussing parametrizations of nonlinear operators with pure boundary interactions.
• Merge the research activities of groups that are currently separately working on spectral theory and existence and uniqueness issues for evolution equations, respectively.
• Enrich the theory of evolution equations on networks by new ideas and variational methods that have been developed in the context of optimal transport over the last two decades.
• Develop new numerical methods for discretization and model reduction of DSNs which preserve important properties of the system and which can be used to analyze and simulate the system.
• Apply the developed theory to real-world engineering problems (e.g., vehicle traffic, water systems, data mining).

Capacity Building Objectives

• Establish an efficient and lasting network of researchers studying DSNs across Europe.
• Link several separate mathematical subfields (from functional analysis to graph theory and numerical analysis) in order to achieve breakthroughs.
• Foster the exchange between ITC and non-ITC researchers, with a special focus on Early Career Investigators.
• Improve gender balance in mathematical research.
• Educate engineers and the professional public on relevant mathematical tools for tackling the investigation of DSNs.

Memorandum of Understanding

Virtual Networking Strategy