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On the coefficient problem and multifractality of whole-plane SLE

Abstract : The starting point of this thesis is Bieberbach’s conjecture: its proof, given by De Branges, uses two ingredients, namely Loewner’s theory of increasing plane domains and an inequality from Milin about the logarithmic coefficients. We start with a study of the logarithmic coefficients of the whole-plane SLE by using a combinatorial method, assisted by computer. We find the results by using a partial differential equation similar to that obtained by Beliaev and Smirnov. We generalize these results by defining the generalized spectrum of the whole-plane SLE, that we calculate by the same method, namely by deriving, thanks to Itô calculus, a parabolic PDE satisfied by the quantities of which we take the average. This two-parameter family of PDEs admits a rich algebraic structure that we study in detail. The last part of this thesis is about the Grunsky operator and its generalizations. In this part that is more experimental we update, thanks to a computer algebra system, a rather complex structure of which we began the exploration.
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Thanh Binh Le. On the coefficient problem and multifractality of whole-plane SLE. General Mathematics [math.GM]. Université d'Orléans, 2016. English. ⟨NNT : 2016ORLE2028⟩. ⟨tel-01477864⟩

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