Forcing in Analysis and Combinatorics

In this thesis, we explore some new applications of the forcing technique in the context of Analysis and Combinatorics by: (1) Constructing a model of Set Theory in which strong measure zero subsets of the real line are meager-additive while Borel’s conjecture fails, answering a long-standing question due to Bartoszy\'nski and Judah. (2) Constructing a model of Set Theory in which Jensen's \diamondsuit_\aleph1 fails, there is a counterexample to Naimark's Problem, and there is a separably represented \textrm{C}^\ast-algebra with exactly two inequivalent irreducible representations. Such a \textrm{C}^\ast-algebra cannot satisfy the conclusion of Glimm's Dichotomy Theorem. (3) Studying families of infinite block sequences of elements of the space \textrm{FIN}_k, Ramsey properties of such families, and Ramsey properties localized on selective or semiselective coideals and ultrafilters.

Department dc:contributor.department

Mathematics

Year dc:date.issued

2024

Author dc:creator

• Calderon Wilches, Daniel

Advisors dc:contributor.advisor

• Todorcevic, Stevo
• Farah, Ilijas

• Combinatorics
• Forcing
• Functional Analysis
• Operator Algebras
• Ramsey Theory
• Real Analysis

Statement dc:rights

• Attribution 4.0 International

Licence dc:rights.uri

• http://creativecommons.org/licenses/by/4.0/

Handle dc:identifier.uri

http://hdl.handle.net/1807/140996

OAI identifier oai:identifier

oai:utoronto.scholaris.ca:1807/140996