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This monograph presents recent and new ideas arising from problems in the field of planar fluid dynamics that are also interesting from the point of view of geometric function theory, potential theory, and two-dimensional stochastic processes. In particular, this book focuses on geometric problems for Hele-Shaw flows and the regularization of singular solutions. Connections to other active branches of mathematics (the Teichmüller space, extremal problems in moduli space of algebraic curves, deformation theory and cohomology of integrable systems) are also discussed. Lastly, the book covers the manifold interactions between several branches of complex and potential analysis, planar fluid mechanics, and differential equations and includes some historical remarks concerning those scientists who have contributed most to the topic. §For most parts of this book, background knowledge provided by graduate courses in real and complex analysis (in particular, the theory of conformal mappings and basic fluid mechanics) is assumed. The book is as self-contained as possible.§
