The classification of objects belonging to some set with respect to a specific property has always been a question of great interest in many areas of mathematics. A classification problem can be formalized as a set \(X\) on which an equivalence relation \(E\) is defined, and the goal is to find a procedure to determine whether two different elements of \(X\) are \(E\)-equivalent or not. We plan to study different classification problems on countable linear orders, proper arcs and knots using the framework of Borel reducibility from Descriptive Set Theory. Borel reducibility was introduced in [FS89] and [HKL90] and has been successfully used in the last thirty years to solve and compare many classification problems. It measures the relative complexity of equivalence relations and it is useful, in the words of Eﬀros, to “classify the unclassifiables”[EFF08]. We are analyzing the sub-interval relation on countable linear orders: even if it is a very natural relation, we found a lower bound for it (with respect to Borel reducibility), which unveils that it is a quite complicated one. Still, we are working to determine its exact complexity. At the same time, we are considering (as an application of the results for such relation) some classification problems on proper arcs and knots.
Moreover, we want to generalize some results of Classical Descriptive Set Theory to Generalized Descriptive Set Theory, a research area which has recently gained popularity, and see some applications to the Braid Theory.
[FS89] Harvey Friedman and Lee Stanley. A Borel reducibility theory for classes of countable structures. J. Symbolic Logic, 54(3):894–914, 1989.
[HKL90] L. A. Harrington, A. S. Kechris, and A. Louveau. A Glimm-Eﬀros dichotomy for Borel equivalence relations. J. Amer. Math. Soc., 3(4):903–928, 1990.
[EFF08] E. G. Eﬀros. Classifying the unclassiﬁables. Contemp. Math. Amer. Math. Soc., 449(369):137–147, 2008.