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This lecture presents Riku Anttila's work on disproving Kleiner's conjecture in geometric analysis. Explore the concept of Q-Loewner spaces, which are metric spaces satisfying Q-Ahlfors regularity and Q-Poincare inequality conditions, and how they enable a rich theory of quasiconformal mappings developed by Heinonen and Koskela. Learn about the combinatorial Loewner property (CLP), a discrete variant of the Loewner property that applies to metric spaces quasisymmetric to Loewner spaces. Discover how Anttila, in joint work with Sylvester Eriksson-Bique, constructed a self-similar CLP-space that is not quasisymmetric to any Loewner space, thereby disproving Kleiner's 2006 conjecture that had suggested all self-similar metric spaces with CLP should be quasisymmetric to Loewner spaces.
