Quantitative embeddings with applications

In this thesis, we discuss quantitative embeddings that generalize a theorem of Kolmogorov and Barzdin. The theorem says that any bounded degree graph with V vertices can be mapped into a 3-dimensional ball of radius sqrt(V), so that at most a constant number of edges intersect any unit ball. In one generalization we describe how much freedom we have in placing the vertices of the graph, and in the other we prove a similar result for simplicial complexes of any dimension. We also discuss applications of these quantitative embeddings to a problem in metric geometry related to the isoperimetric inequality and a problem about constructing local quantum error-correcting codes.

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

2025

Author dc:creator

• Portnoy, Elia

Advisor dc:contributor.advisor

• Guth, Lawrence

Statement dc:rights

• Attribution-NonCommercial-NoDerivatives 4.0 International (CC BY-NC-ND 4.0)
• Copyright retained by author(s)

Licence dc:rights.uri

• https://creativecommons.org/licenses/by-nc-nd/4.0/

Handle dc:identifier.uri

https://hdl.handle.net/1721.1/159936

OAI identifier oai:identifier

oai:dspace.mit.edu:1721.1/159936