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BIMSA-Tsinghua Quantum Symmetry Seminar
Noncommutative Geometry on the Berkovich projective line
Noncommutative Geometry on the Berkovich projective line
Organizers
Speaker
Damien Tageddine
Time
Wednesday, May 28, 2025 10:30 AM - 12:00 PM
Venue
A3-3-301
Online
Zoom 242 742 6089
(BIMSA)
Abstract
The Berkovich projective line is an analytic space over a non-Archimedean field. Over the field of complex $p$-adic numbers Cp, it can be realized as an infinite R-tree with
a dense number of branching points and countable branches at each of these points. The automorphism group of the R-tree is in this case identified with PGL(2,Cp).
In this talk, we will review the representation theory of groups acting on R-trees. The theory is well understood in the case of locally compact trees. Interesting results can also be derived in
the case of R-trees with countable branches such as the Berkovich line.
Using this machinery, we exhibit a cross-product C*-algebra on the Berkovich line and construct an unbounded spectral triple on this space. This allows us to develop a noncommutative
harmonic analysis on the Berkovich projective line over Cp. We show that invariant measures, such as the Patterson-Sullivan measure, can be obtained as KMS-states of this C*-algebra.
This is a joint work with Masoud Khalkhali
Reference: https://arxiv.org/abs/2411.02593
a dense number of branching points and countable branches at each of these points. The automorphism group of the R-tree is in this case identified with PGL(2,Cp).
In this talk, we will review the representation theory of groups acting on R-trees. The theory is well understood in the case of locally compact trees. Interesting results can also be derived in
the case of R-trees with countable branches such as the Berkovich line.
Using this machinery, we exhibit a cross-product C*-algebra on the Berkovich line and construct an unbounded spectral triple on this space. This allows us to develop a noncommutative
harmonic analysis on the Berkovich projective line over Cp. We show that invariant measures, such as the Patterson-Sullivan measure, can be obtained as KMS-states of this C*-algebra.
This is a joint work with Masoud Khalkhali
Reference: https://arxiv.org/abs/2411.02593
Speaker Intro
Damien Tageddine (pronounced Ta:jédeen) has completed his PhD thesis at McGill University in 2024. He has been collaborating with Masoud Khalkhali (Western Ontario) since 2023. His main research interest is on noncommutative geometry; specially his work focuses on spaces closely tied to discrete structures (such as discrete groups, trees, R-trees, graphs etc…).