## Departmental PhD Thesis Exam – Francisco Guevara Parra

Everyone is welcome to attend.  Refreshments will be served in the Math Lounge before the exam.

Friday, March 15, 2019
1:30 p.m.
Fields Institute, room 210

PhD Candidate:  Francisco Guevara Parra
Supervisor:   Stevo Todorcevic
Thesis title:  Analytic spaces and their Tukey types

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In this Thesis we study topologies on countable sets from the perspective of Tukey reductions of their neighbourhood filters. It turns out that is closely related to the already established theory of definable (and in particular analytic) topologies on countable sets. The connection is in fact natural as the neighbourhood filters of points in such spaces are typical examples of directed sets for which Tukey theory was introduced some eighty years ago. What is interesting here is that the abstract Tukey reduction of a neighbourhood filter $\mathcal{F}_{x}$ of a point to standard directed sets like $\mathbb{N}^\mathbb{N}$ or $\ell_1$ imposes that $\mathcal{F}_{x}$ must be analytic. We develop a theory that examines the Tukey types of analytic topologies and compare it by the theory of sequential convergence in arbitrary countable topological spaces either using forcing extensions or axioms such as, for example, the Open Graph Axiom. It turns out that in certain classes of countable analytic groups we can classify all possible Tukey types of the corresponding neighbourhood filters of identities. For example we show that if $G$ is a countable analytic $k$-group then $1=\{0\},$ $\mathbb{N}$ and $\mathbb{N}^\mathbb{N}$ are the only possible Tukey types of the neighbourhood filter $\mathcal{F}_{e}^{G}$. This will give us also new metrization criteria for such groups. We also show that the study of definable topologies on countable index sets has natural analogues in the study of arbitrary topologies on countable sets in certain forcing extensions.

A copy of the thesis can be found here: Francisco_PhD_Thesis