PhD Defense Severin Mejak
Definability of maximal discrete sets
Abstract:
Abstract In this thesis we study set-theoretic definability of maximal objects originating from various branches of mathematics, encompassing set theory, combinatorics, group theory, measure theory and operator algebras. In Part II, which is based on joint work with Asger Törnquist, we study definability of maximal almost disjoint families. With a simple tree derivative process, we first give a new proof of the classical theorem, due to Mathias, stating that there are no infinite analytic maximal almost disjoint families. With small adjustments, the process can be carried out and terminates in Lω CK 1 , which proves that for every infinite Σ 1 1 almost disjoint family A there is a ∆1 1 infinite subset x of ω such that x ∩ z is finite for every z ∈ A. Our argument can be adapted to prove that if ℵ L[a] 1 < ℵ1, then there are no infinite Σ 1 2 [a] maximal almost disjoint families. A small modification of the derivative process can also be used to prove that under MA(κ) there are no infinite κ-Suslin maximal almost disjoint families. Part III is a reproduction of a preprint on definability of maximal cofinitary groups, authored jointly with David Schrittesser. We give a construction of a closed (even Π0 1 ) set which freely generates an Fσ (even Σ 0 2 ) maximal cofinitary group. In this isomorphism class, this is the lowest possible complexity of a maximal cofinitary group. Additionally, we discuss obstructions to potential constructions of Gδ maximal cofinitary groups and introduce (maximal) finitely periodic groups. In Part IV, which is also a reproduction of a preprint, we study maximal ortho[1]gonal families. We begin by giving a new, short and elementary proof of a theorem by Preiss and Rataj, stating that there are no analytic maximal orthogonal families of Borel probability measures on a Polish space. In case when the underlying space is compact and perfect, we establish that the set of witnesses to non-maximality is comeagre. The idea of our argument is based on the original proof by Preiss and Rataj, but with significant simplifications. Our proof generalises to show that under MA + ¬CH there are no Σ1 2 maximal orthogonal families, that under PD there are no projective maximal orthogonal families and that under AD there are no maximal orthogonal families at all. Finally, we introduce a notion of strong orthogonality for states on separable C*-algebras and generalise a theorem due to Kechris and Sofronidis, stating that for every analytic orthogonal family of Borel probability measures there is a product measure orthogonal to all measures in the family, to states on a certain class of C*-algebras. ii
Hybrid Defense :
https://ucph-ku.zoom.us/j/64607910696
Meeting ID: 646 0791 0696
Advisors: Asger Törnquist, University of Copenhagen, Denmark
David Schrittesser, Harbin Institute of Technology, China and University of Toronto, Canada
Assessment committee:
Ryszard Nest (chair), University of Copenhagen, Denmark
Alessandro Andretta, University of Turin, Italy
Vera Fischer, University of Vienna, Austria