| Résume | The notion of weak embeddability between ordered structures dates back to the early post-Cantor era (e.g. in a 1915 paper by Hartogs,) possibly in a different terminology. The notion has attracted attention in various mathematical contexts after it was explored and popularised by Todorcevic, Mekler, and Väänänen.
In this talk, I will discuss the concept of weak embeddability between ordered structures and highlight its historical significance and mathematical importance. I will demonstrate why and how weak embeddability, especially between uncountable trees, naturally appears in and builds bridges between various fields, including universality problems, model theory, partition properties and generalised descriptive set theory. I will mention recent advances in the famous problem of having a universal tree in the class of Aronszajn trees. If time permits, I will also present a proof sketch showing that the universality number of Aronszajn trees is never the first uncountable cardinal number. |
