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Quasiconformal mappings form a natural extension of conformal mappings and play a fundamental role in modern complex analysis, geometry, and low-dimensional topology. This lecture series will provide an introduction to the theory of quasiconformal mappings, emphasizing the geometric ideas that lead naturally to Teichmüller theory. After developing the basic theory of quasiconformal mappings and their relationship with complex structures on Riemann surfaces, we shall construct Teichmüller space using quasiconformal deformations and discuss Teichmüller mappings and their extremal properties. We will then introduce the complex analytic structure of Teichmüller space through the Bers embedding. If time permits, the series will conclude with a proof of Royden's celebrated theorem, which identifies the Teichmüller metric with the Kobayashi metric, illustrating the deep connections between quasiconformal mappings, intrinsic complex geometry, and the theory of moduli spaces. |