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Final Report Summary - STREVCOMS (Structure and Evolution of Complex Systems with Applications in Physics and Life Sciences)

The aim of the project is to make a major step forward in the theory of broadly understood complex systems based on direct considering their constituents and deducing the laws that govern the behavior of such a system as a whole from the individual interactions between its constituents. The results obtained by the consortium range from developing and studying individual-based models of randomly evolving shapes and spreading infectious diseases to elaborating new effective cryptographic systems or opening new horizons in the theory of quantum matter. Among the results are also new facts and concepts in the theory of holomorphic maps in Banach spaces, the explanation of percolation-related phenomena in liquid crystals or the description of the appearance of photo-controllable networks of nanoparticles, as well as elaborating a mathematically rigorous and practically useful framework for individual-based modeling of large populations of interacting species. These results essentially extend understanding the behavior and properties of complex real-world objects and phenomena far beyond the usual macroscopic (heuristic) level. Such results as modeling: (a) dynamics of infectious diseases and other communities of microscopic species; (b) percolation-related phenomena in liquid crystals; (c) the flow of the enveloped drug agents through blood vessels; (d) appearance of photo-controllable networks of nanoparticles, may have direct applications and hence can serve as the base of new research projects in these directions.
All our tasks and objectives were achieved, some of them far beyond the frontiers that we had outlined at the beginning. During the four-year period of our work on the project, the obtained research results were published in 115 peer reviewed articles and book chapters. Another 47 articles were submitted for publication, many of them have already been accepted. From the perspective of our application-related work packages, we achieved the following.
Complex network: theory, simulations, applications: Studying properties of multicomponent molecular networks/gels with applications to the theory of liquid crystals. This includes elaborating and studying by computer simulation methods coarse-grained models describing the dynamics of formation and equilibrium properties of multicomponent gels. Explaining the universal shape characteristics for mesoscopic polymer chains immersed in different solvents. Contributing to the theory of star-like polymers in solvents and micelle formation. Contributing to the theory of irregular and random graphs and networks by studying their properties and also those of Gibbs random fields on such structures. Developing a complex-network approach to visualizing and quantifying the evolution of scientific topics. Elaborating new cryptographic systems based on the idea of hidden discrete logarithms. Contributing to the theory of Gibbs states of classical models on random geometric graphs, including the theory of liquid-vapor phase transition. Developing the theory of quantum systems with deformed commutation relations and with non-commutative particle coordinates.
Microscopic modeling in spatial and evolutionary biology: Elaborating a general mathematically rigorous and practically useful framework that might be widely applicable in individual-based modeling of animal movement and in other problems of theoretical ecology. In particular, studying the spatial logistic model and its generalization towards including dispersion spreading and monostable reactions. Developing and studying individual-based models of disease spread based on random graphs. For these models, general conclusions were drawn towards individual-based modeling of the spread of infections and similar phenomena in random environment.
Microscopic (Markov) dynamics of continuum particle systems with applications: Developing the dynamical theory of infinite systems of point entities in the continuum with generators of various types, including those describing jumps with attraction and repulsion, coagulation, fragmentation, birth-and-death processes, multi-type particle systems, etc. In particular, the dynamics of a spatial ecological model was described in which the constituent entities reproduce themselves at distant points (disperse) and die with rate that includes a competition term. The role of the tradeoff between the dispersal and competition in such systems was revealed. The Markov evolution of states of a continuum migration model was also described. A general approach to studying interacting particle systems was developed that employs considering the system at different hierarchical levels. The basics of the p-adic analysis on configuration spaces and a p-adic spatial combinatorics were elaborated.
Compartmental models with time delays and stochasticity: A general mathematical framework for individual-based models (IBMs) containing interactions of an unlimited level of complexity was elaborated and equations that reliably approximate the effects of space and stochasticity were derived. We provided software, specified in an accessible and intuitive graphical way, so that any researcher can obtain analytic and simulation results for any particular IBM. We illustrated the framework with examples from conservation biology and evolutionary ecology. A number of realistic IBMs describing infectious disease spread were studied. This includes models based on massive datasets obtained recently during pandemic spread of some viral diseases.
Mathematics and physics of the evolution of complex shapes: An approach was elaborated that allows for constructing the evolution of complex shapes in terms of new processes of Stochastic Löwner Evolution (SLE) type that possess conformal invariance and the domain Markov property. A version of the conformal field theory was developed based on the background charge and Dirichlet boundary condition modifications of Gaussian Free Field. Numerical simulation of the general SLE was developed and implemented by using Wolfram Mathematica software.
All our tasks on networking and training young researchers were achieved. 53 team members participated in the project related secondments, which allowed them – that is especially important for early stage researchers – to work side-by-side with renowned scientists involved in the consortium. In 2016, two PhD dissertations were completed and defended by our early state researchers. Another seven such works – 4 in Lublin, 1 in Temuco and 2 in Lviv – are almost finished and will be defended within the next year.

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Life Sciences
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