Frontiers of Liouville quantum gravity

Abstract This thesis studies a universal model of random geometry in two dimensions called Liouville quantum gravity (LQG). LQG was originally described heuristically by physicists, and mathematicians have grappled with the challenge of defining it rigorously and analyzing its properties. We investigate elements of the theory of LQG that are still poorly understood, often even from physicists' heuristic perspective. -- We analyze LQG as a metric space. We prove results necessary for the construction of LQG as a metric space, and prove fundamental estimates for these distances. We prove the most natural formulation of the Knizhnik-Polyakov-Zamolodchikov (KPZ) formula, which relates Hausdorff dimensions of sets with respect to the Euclidean and LQG metric. And we prove upper and lower bounds on the Hausdorff dimension of the LQG metric. --

Name thesis:degree_name

Doctoral

Department dc:contributor.department

Massachusetts Institute of Technology. Department of Mathematics

Grantor dc:publisher

Massachusetts Institute of Technology

Year dc:date.issued

2020

Author dc:creator

• Pfeffer, Joshua William.

Advisor dc:contributor.advisor

• Scott Sheffield.

• Mathematics.

Statement dc:rights

• MIT theses may be protected by copyright. Please reuse MIT thesis content according to the MIT Libraries Permissions Policy, which is available through the URL provided.

Licence dc:rights.uri

• http://dspace.mit.edu/handle/1721.1/7582

Language dc:language.iso

eng

Handle dc:identifier.uri

https://hdl.handle.net/1721.1/126933

OAI identifier oai:identifier

oai:dspace.mit.edu:1721.1/126933