Logic Seminar: Tan Özalp - University of Notre Dame

Speaker: Tan Ozalp
University of Notre Dame

Will give a Logic Seminar entitled:
Initial Tukey Structure Below a Stable Ordered-Union Ultrafilter

Abstract: Tukey introduced the notion of Tukey ordering for partial orders to study convergence in general topological spaces. The Tukey ordering of a particular class of partial orders, namely of the class of ultrafilters has been a major topic of interest in the last 16+ years. Since Todorcevic’s (2012) proof that Ramsey ultrafilters are Tukey minimal, the investigation of initial Tukey structures below special ultrafilters has been an important part of this work. Subsequently, the initial Tukey structures below ultrafilters forced by Laflamme's partial orders were classified by Dobrinen and Todorcevic (2014 and 2015), below ultrafilters forced by $\mathcal{P}(\omega^k)/{\mathrm{Fin}}^{\otimes k}$ for all $k \geq 2$ were classified by Dobrinen (2016), and below ultrafilters associated to topological Ramsey spaces constructed from Fraïssé classes were classified by Dobrinen, Mijares and Trujillo (2017). The classification of the initial Tukey structure below a stable ordered-union ultrafilter was a question of the 2011 paper of Dobrinen and Todorcevic, which also appeared in Kuzeljevic and Raghavan's survey paper (2024). Continuing this line of work, we prove that there are exactly 5 Tukey classes below a stable ordered-union ultrafilter. For this, we first simplify a canonization theorem of Klein and Spinas (2005) for Borel functions on the Milliken space of infinite block sequences to the context of fronts on this space, and then we utilize this in the proof of the initial Tukey structure result.

Date: 11-05-2024
Time: 2:00 pm
Location: 125 Hayes-Healy Bldg

Download Poster [PDF, 209k]

Originally published at math.nd.edu.