Programma Giovani Ricercatori "Rita Levi Montalcini"
New advances in Descriptive Set Theory
Duration: 3 years (starting September 2014)
Principal Investigator: Luca Motto Ros
This project focuses on Descriptive Set Theory (briefly: DST), one of the most stimulating and well-established branches of mathematical logic, whose results and methods have been successfully applied to various areas of mathematics (such as representation theory for topological groups, ergodic theory, classical analysis, Banach spaces and operator algebras, to name a few). DST is primarily concerned with the study of definable subsets of the real line or, more generally, of Polish spaces. Such sets are usually classified in suitable hierarchies of complexity, with the purpose of giving a global description of such hierarchies, of systematically analyzing their levels, and of determining the exact position in them of particularly relevant mathematical objects. This kind of analysis is strongly connected with the problem of classifying mathematical objects in a given class (providing methods and invariants to obtain it, or showing that a classification is unfeasible), a widespread theme in mathematics. With this project, we aim to reach relevant advances on some mainstream themes of modern DST. In particular, we plan to tackle the following main research topics:
A) Wadge-like hierarchies on arbitrary topological spaces
The abstract analysis of the Wadge hierarchy (which gives the finest possible notion of topological complexity) on zero-dimensional Polish spaces had a remarkable impact on set theory and theoretical computer science. In this project, we will extend this analysis to Wadge-like hierarchies on other kind of spaces, in particular on topologically complex spaces like omega-continuous domains.
B) Decomposable Borel functions
One of the classical themes in DST is the study of the structure of Borel functions, in particular with respect to their decomposability into countably-many continuous functions. Exploiting some recent progress in this area, we aim to obtain a generalization to all finite levels of a deep theorem of Jayne and Rogers, solving in this way an important problem which remained open over the last 30 years.
C) Invariantly universal analytic quasi-orders
In the last 20 years, the classification of definable quasi-orders and equivalence relations by suitable reducibility notions has been one of the mainstream topics in DST. In this direction, we will study a recently emerged property called invariant universality, investigate the mutual relationship among the variants of the classical notion of Borel reducibility which have appeared in the literature, and examine the interplay between DST and various relevant notions of forcing.
D) Generalized DST
One of the most active and fast-growing branches of DST is the study of the generalized Baire and Cantor spaces. In our project, we plan to use DST methods to analyze the topological complexity of the embeddability relation between uncountable structures (which may be construed as an analytic quasi-order on the generalized Cantor space).