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Bost–Connes–Marcolli System for the Siegel Modular Variety


Abouamal, Ismail (2024) Bost–Connes–Marcolli System for the Siegel Modular Variety. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/hjp3-8a71.


Bost–Connes–Marcolli systems are an important type of quantum statistical mechanical systems which provide connections between the ergodic theory of dynamical systems, class field theory and von Neumann algebras. In this thesis, we generalize the GL₂-system to the case of the Siegel modular variety of degree two and study its various properties. We show that this dynamical system undergoes a spontaneous symmetry breaking phase transition and classify its equilibrium extremal states at different inverse temperatures. We next study its symmetry group and derive an intertwining equality between the action by symmetries and a subgroup of the Galois group of the Siegel modular field. Finally, we study the von Neumann algebras associated to the equilibrium states and classify their types.

Item Type:Thesis (Dissertation (Ph.D.))
Subject Keywords:quantum statistical mechanics; C* dynamical systems; von-Neumann algebras; Bost-Connes system; GL2 system; symmetry breaking; class field theory; ergodic theory;
Degree Grantor:California Institute of Technology
Division:Physics, Mathematics and Astronomy
Major Option:Mathematics
Awards:Scott Russell Johnson Graduate Dissertation Prize in Mathematics, 2024.
Thesis Availability:Public (worldwide access)
Research Advisor(s):
  • Marcolli, Matilde
Thesis Committee:
  • Mantovan, Elena (chair)
  • Kechris, Alexander S.
  • Khalkhali, Masoud
  • Marcolli, Matilde
Defense Date:17 May 2024
Funding AgencyGrant Number
Natural Sciences and Engineering Research Council of Canada (NSERC)PGSD 2-535022- 2019
Record Number:CaltechTHESIS:06032024-064737346
Persistent URL:
Abouamal, Ismail0009-0004-0462-8177
Default Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:16477
Deposited By: Ismail Abouamal
Deposited On:03 Jun 2024 23:46
Last Modified:17 Jun 2024 19:32

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