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Mathematics of the Holographic Principle

Citation

Gesteau, Elliott M. (2025) Mathematics of the Holographic Principle. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/2mkb-nz86. https://resolver.caltech.edu/CaltechTHESIS:06012025-234156081

Abstract

The holographic principle, which states that quantum gravity in a given spacetime region admits an equivalent description in terms of a quantum system without gravity on its boundary, is a very promising candidate to lay the foundations of our understanding of quantum gravity. However, a precise general formulation of this principle, as well as its domain of applicability, are yet to be understood. This thesis explores the foundations of holography from a mathematical point of view. In particular, the theory of von Neumann algebras is exploited to understand features of the emergence of spacetime in holography, leveraging tools from quantum error correction in infinite dimensions as well as results on harmonic analysis. Some connections between hyperbolic geometry and recent developments in holography are also elucidated, an approach to the factorization of holographic theories based on Hopf algebras is developed, and a puzzle regarding the description of a closed universe within the context of the AdS/CFT correspondence is put forward.

Item Type:Thesis (Dissertation (Ph.D.))
Subject Keywords:Holography; Quantum gravity; von Neumann algebras; Large N limit; Harmonic analysis; Hyperbolic geometry; Quantum error correction; Hopf algebras; Tensor networks; Bootstrap; Ergodic hierarchy; Topology change
Degree Grantor:California Institute of Technology
Division:Physics, Mathematics and Astronomy
Major Option:Mathematics
Awards:Scott Russell Johnson Graduate Dissertation Prize in Mathematics, 2025. Scott Russell Johnson Prize for Excellence in Graduate Studies, 2023. Scott Russell Johnson Prize for Excellence as a First-Year Graduate Student, 2021.
Thesis Availability:Public (worldwide access)
Research Advisor(s):
  • Marcolli, Matilde
Thesis Committee:
  • Ooguri, Hirosi (chair)
  • Marcolli, Matilde
  • Kapustin, Anton N.
  • Liu, Hong
Defense Date:27 May 2025
Record Number:CaltechTHESIS:06012025-234156081
Persistent URL:https://resolver.caltech.edu/CaltechTHESIS:06012025-234156081
DOI:10.7907/2mkb-nz86
Related URLs:
URLURL TypeDescription
https://doi.org/10.1007/s00220-024-05192-3DOIArticle adapted for Ch.2
https://doi.org/10.1103/PhysRevD.110.106005DOIArticle adapted for Ch.3
https://doi.org/10.48550/arXiv.2408.12642DOIArticle adapted for Ch.4
https://doi.org/10.4310/ATMP.241031230051DOIArticle adapted for Ch.5
https://doi.org/10.56994/JAMR.003.001.003DOIArticle adapted for Ch.6
https://doi.org/10.1007/JHEP10(2022)169DOIArticle adapted for Ch.7
https://doi.org/10.48550/arXiv.2504.00096DOIArticle adapted for Ch.8
https://doi.org/10.1088/1751-8121/acef7dDOIArticle adapted for Ch.9
https://doi.org/10.48550/arXiv.2407.20324DOIArticle adapted for Ch.10
https://doi.org/10.48550/arXiv.2504.14586DOIArticle adapted for Ch.11
ORCID:
AuthorORCID
Gesteau, Elliott M.0000-0002-3189-0953
Default Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:17348
Collection:CaltechTHESIS
Deposited By: Elliott Gesteau
Deposited On:06 Jun 2025 20:33
Last Modified:17 Jun 2025 18:36

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