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Explore the noncommutative geometry of the Berkovich projective line in this lecture by Damien Tageddine from McGill University. Delve into how the Berkovich projective line over the field of complex p-adic numbers Cp can be realized as an infinite R-tree with dense branching points and countable branches, with its automorphism group identified as PGL(2,Cp). Learn about representation theory of groups acting on R-trees, the construction of a cross-product C*-algebra on the Berkovich line, and the development of an unbounded spectral triple. Discover how this framework enables noncommutative harmonic analysis and how invariant measures like the Patterson-Sullivan measure can be obtained as KMS-states of the C*-algebra. The talk is based on joint work with Masoud Khalkhali and references the paper available at https://arxiv.org/abs/2411.02593.
