Boundaries of a Complex WorldThe central theme of this book is the extent to which the structure of the free dynamical boundaries of a system controls the evolution of the system as a whole. Applying three orthogonal types of thinking  mathematical, constructivist and morphological, it illustrates these concepts using applications to selected problems from the social and life sciences, as well as economics. In a broader context, it introduces and reviews some modern mathematical approaches to the science of complex systems. Standard modeling approaches (based on nonlinear differential equations, dynamic systems, graph theory, cellular automata, stochastic processes, or information theory) are suitable for studying local problems. However they cannot simultaneously take into account all the different facets and phenomena of a complex system, and new approaches are required to solve the challenging problem of correlations between phenomena at different levels and hierarchies, their selforganization and memoryevolutive aspects, the growth of additional structures and are ultimately required to explain why and how such complex systems can display both robustness and flexibility. This graduatelevel text also addresses a broader interdisciplinary audience, keeping the mathematical level essentially uniform throughout the book, and involving only basic elements from calculus, algebra, geometry and systems theory.

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What is fascinating about this book is the fact that just a simple component of Complexity namely the "boundary" can be extremely complex. So, complexity appears even in the components of Complexity itself.
Even from the first chapter, which is extremely beautiful called ``Nonlineart'', it is shown that in the visual arts just simple frames can produce a lot of different perceptions (to mention multispatiality, temporal dependence etc.) presented extremely nice and clear, showing an accurate artistic and scientific insight. Even though this book is a scientific one, the first chapter blends unexpectedly scientific and artistic views, rarely found in literature.
Then the author starts to present the most nonlinear world, namely the boundary dynamics of social systems, which today represents a very deep and painful problem. Class, ethnic and gender inequality, spatial or spiritual boundaries of communities and national identities, reveal the fact that managing just the frames/boundaries can help tremendously in understanding and control the complex dynamics involved. Furthermore it can help in answering to big questions which appear whenever one studies societies, namely ``what is the way in which long lasting social structures appear and survive out of social interactions?''. In addition, seems that social boundaries are viewed as a way to ``parametrize'' the relationality in a way parallel to the case of cellular biology in which cell membranes are crucial to the whole synergetics of any tissue. Mathematical approaches to ``social distances'' are presented together with various models of growth, cooperation (in the rigorous sense of replicator dynamics) networking and pattern formation.
Next two chapters the book are devoted on mathematical tools needed to understand the boundary dynamics. First, the differential geometric/topologic approach is presented extremely clear (extremely intuitive but rigorous as well) starting from very simple aspects about vector fields, surfaces, connections, to highly nontrivial topological aspects like vector bundles and cobordism. Second part presents the other face of the coin, discrete mathematics which is the main tool in studying the evolutionary dynamics of ``populations'' in a very general sense. Starting with the elements of graph theory and ending with elements of algebraic topology (including triangulations, homology groups, homotopy, etc.)
Finally, the book discusses many applications from internet networks to crazy mathematics of soap films, fluids and nonlinear selforganised waves propagating in nerves.
The pivotal role in the whole book is played by the boundary, as the key element which makes separation among various components. Inasmuch as the complexity itself is a ``collection'' of a huge variety of components with a huge variety of interactions, one can see immediately why this book is special in the whole literature discussing complexity and emergence.
Dr. Adrian Stefan Carstea,
Research Group of Geometry and Physics
(GAP) IFINHH, Bucharest, 2016