Alberto De Castelli

Ph.D XL

Supervisor: Vincenzo Dimonte

Phone: 

Room: Rizzi L2-07-ND

Mail: decastelli.alberto@spes.uniud.it

Research Project

GDST of Higher Polish Transformation Groups at Singular Cardinals

A Polish group is a topological group whose topology is completely metrizable and separable. In classical descriptive set theory, there are a lot of interesting results concerning this kind of spaces and their continuous (or Borel) actions on Polish spaces. In our framework, we generalize this definition and consider all those completely metrizable topological groups which admit a basis of at most lambda-many open sets, where lambda is a singular cardinal of countable cofinality. Working in this setting, it's possible to obtain a new theory which is of course weaker than the classical one, but still solid.

Cardinal Invariants of the Ellentuck-Prikry Topology at Singular Cardinals

One of the most important failures of GDST at singular cardinals is the higher Baire Category Theorem: the higher Baire space lambda^omega endowed with the bounded topology admits aleph_1-many open dense sets whose intersection is empty. However, using a suitable forcing notion, there is another topology (called Ellentuck-Prikry topology) which refines the previous one: on one hand we lose metrizability, but on the other we recover a lot of higher Category arguments like the higher Baire Category Theorem, the Kuratowski-Ulam theorem and the Mycielski's independence theorem. We are interested in studying the cardinal invariants of such a topology, investigating which configuration of the higher Cichòn's diagram can be realized using a generic extension.

References:

  • Su Gao. Invariant Descriptive Set Theory.
  • Howard Becker, Alexander S.Kechris. Descriptive Set Theory of Polish Group Actions.
  • Vincenzo Dimonte, Luca Motto Ros. Generalized Descriptive Set Theory at Singular Cardinals with Countable Cofinality.
  • Vincenzo Dimonte, Martina Iannella, Philip Lucke. Descriptive Properties of I2 embeddings.
  • Moti Gitik. Prikry-Type Forcings. Handbook of Set Theory.
  • Andreas Blass. Combinatorial Cardinal Characteristics of the Continuum.