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Over the last fifteen years fractal geometry has established itself as a substantial mathematical theory in its own right. The interplay between fractal geometry, analysis and stochastics has highly influenced recent developments in mathematical modeling of complicated structures. This process has been forced by problems in these areas related to applications in statistical physics, biomathematics and finance. This book is a collection of survey articles covering many of the most recent developments, like Schramm-Loewner evolution, fractal scaling limits, exceptional sets for percolation, and heat kernels on fractals. The authors were the keynote speakers at the conference "Fractal Geometry and Stochastics IV" at Greifswald in September 2008.

Fractals. --- Stochastic processes. --- Fractals --- Stochastic processes --- Mathematics --- Geometry --- Mathematical Statistics --- Physical Sciences & Mathematics --- Random processes --- Fractal geometry --- Fractal sets --- Geometry, Fractal --- Sets, Fractal --- Sets of fractional dimension --- Mathematics. --- Geometry. --- Probabilities. --- Probability Theory and Stochastic Processes. --- Probabilities --- Dimension theory (Topology) --- Distribution (Probability theory. --- Distribution functions --- Frequency distribution --- Characteristic functions --- Euclid's Elements --- Probability --- Statistical inference --- Combinations --- Chance --- Least squares --- Mathematical statistics --- Risk

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Functional analysis --- Measure theory. --- Fractals. --- Self-similar processes. --- Mesure, Théorie de la --- Fractales --- Processus autosimilaires --- 51 <082.1> --- Mathematics--Series --- Fractales. --- Mesure, Théorie de la. --- Processus autosimilaires. --- Mesure, Théorie de la --- Fractals --- Measure theory --- Self-similar processes --- Selfsimilar processes --- Stochastic processes --- Fractal geometry --- Fractal sets --- Geometry, Fractal --- Sets, Fractal --- Sets of fractional dimension --- Dimension theory (Topology) --- Lebesgue measure --- Measurable sets --- Measure of a set --- Algebraic topology --- Integrals, Generalized --- Measure algebras --- Rings (Algebra)

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For metric spaces the quest for universal spaces in dimension theory spanned approximately a century of mathematical research. The history breaks naturally into two periods — the classical (separable metric) and the modern (not necessarily separable metric). While the classical theory is now well documented in several books, this is the first book to unify the modern theory (1960 – 2007). Like the classical theory, the modern theory fundamentally involves the unit interval. By the 1970s, the author of this monograph generalized Cantor’s 1883 construction (identify adjacent-endpoints in Cantor’s set) of the unit interval, obtaining — for any given weight — a one-dimensional metric space that contains rationals and irrationals as counterparts to those in the unit interval. Following the development of fractal geometry during the 1980s, these new spaces turned out to be the first examples of attractors of infinite iterated function systems — “generalized fractals.” The use of graphics to illustrate the fractal view of these spaces is a unique feature of this monograph. In addition, this book provides historical context for related research that includes imbedding theorems, graph theory, and closed imbeddings. This monograph will be useful to topologists, to mathematicians working in fractal geometry, and to historians of mathematics. It can also serve as a text for graduate seminars or self-study — the interested reader will find many relevant open problems that will motivate further research into these topics.

Dimension theory (Topology). --- Ethnomathematics. --- Fractals. --- Mathematics -- Social aspects. --- Fractals --- Dimension theory (Topology) --- Mathematics --- Physical Sciences & Mathematics --- Geometry --- Fractal geometry --- Fractal sets --- Geometry, Fractal --- Sets, Fractal --- Sets of fractional dimension --- Mathematics. --- Mathematical analysis. --- Analysis (Mathematics). --- Dynamics. --- Ergodic theory. --- Functions of complex variables. --- Topology. --- Analysis. --- Functions of a Complex Variable. --- Dynamical Systems and Ergodic Theory. --- Analysis situs --- Position analysis --- Rubber-sheet geometry --- Polyhedra --- Set theory --- Algebras, Linear --- Complex variables --- Elliptic functions --- Functions of real variables --- Ergodic transformations --- Continuous groups --- Mathematical physics --- Measure theory --- Transformations (Mathematics) --- Dynamical systems --- Kinetics --- Mechanics, Analytic --- Force and energy --- Mechanics --- Physics --- Statics --- 517.1 Mathematical analysis --- Mathematical analysis --- Math --- Science --- Topology --- Global analysis (Mathematics). --- Differentiable dynamical systems. --- Differential dynamical systems --- Dynamical systems, Differentiable --- Dynamics, Differentiable --- Differential equations --- Global analysis (Mathematics) --- Topological dynamics --- Analysis, Global (Mathematics) --- Differential topology --- Functions of complex variables --- Geometry, Algebraic