Ivanyos János Balázs: Forcing methods in set theory and topology

The proposed research investigates fundamental problems at the intersection of set theory and topology, with particular emphasis on the use of forcing methods. The project will focus on the role of special families of sets, such as $Q$-sets and $\Delta$-sets, as well as on cardinal spectra of cardinal invariants, using them as case studies to better understand the relationship between combinatorial set-theoretic principles and topological invariants.

More broadly, the aim is to analyze how forcing constructions (e.g., Cohen, Sacks, or iterated forcing) can separate or collapse different topological spaces or classes of sets of reals, and to determine which cardinal invariants govern these phenomena. Sample guiding problems: (1) Is it consistent that there exists a $\Delta$-set which is not a $Q$-set? (2) Does the wFN property of $\aleph_\omega$ imply the existence of a splendid space of size $\aleph_{\omega+1}$ under GCH? (3) Characterize the cardinal spectrum of the cardinal invariant $\mathfrak{a}$.