Tukey reduction among analytic directed orders

TL;DR

This survey reviews recent developments in the structure of Tukey reductions among analytic ideals, organizing results to highlight unresolved problems and including new joint findings with Stevo Todorcevic.

Contribution

It provides a unified organization of recent results on Tukey reductions among analytic ideals and introduces new joint findings.

Findings

Identification of unresolved problems in the field

Organization of results into a cohesive framework

Announcement of new joint results with Stevo Todorcevic

Abstract

This is a survey of recent work on the structure of Tukey reductions among analytic $\sigma$-ideals of compact subsets of compact metric spaces and analytic P-ideals of sets of natural numbers. An attempt is made to organize the results into a unified whole. This organization makes it possible to identify natural unresolved problems. Some new joint results with Stevo Todorcevic are announced.