Title: Uncountably many real dimensions of typesover Hrushovski’s omega-categorical pseudoplane
Abstract
In 1974, Lachlan conjectured that any omega-categorical stable structure isomega-stable.In 1989 E. Hrushovski refuted Lachlan’s conjecture by using an amalgamationconstruction, obtaining an omega-categorical and strictly stable pseudo plane. Theproof used by Hrushovski to refute the conjecture was indirect and used a factproven by Cherlin, Harington, and Lachlan in 1985: There are no omega-categorical and omega-stable pseudoplanes. In this lecture, we will review my master thesis, where we give direct proof for that property by producing a real-valued function over the space of types, invariant under isomorphisms, with an uncountable image. Moreover, as a by-product, we alsorefute a conjecture by Krupiński that any omega-categorical, small NIP structureinduces a small Polish structure.