Ph.D XL

Supervisor: Vincenzo Dimonte

Phone: 

Room: RIZ1 - L2-07-ND

Mail: decastelli.alberto@spes.uniud.it

Research Project

GDST of λ-Polish Group Actions

My work intends to bring some understanding about different features of Generalized Polish Group Theory, like Polish group actions, amenability and Borel reducibility. In order to investigate these properties, we use different tools from the world of set theory: for example, we will use a lot of results from Generalized Descriptive Set Theory and General Topology. It turns out that, in order to have a theory which is symmetric with the classical one, we may need to work with very large cardinals, and more precisely with Woodin cardinals. Moreover, we often have to change topology in order to recover some properties, so we need to use different types of Forcing (Prikry Forcing, Cohen Forcing and so on).

Let λ be a strong limit cardinal with countable cofinality. We want to investigate the following definition: a λ-Polish group is a topological group which is completely metrizable and has weigth less than or equal to λ, i.e. there exists a topological (wellordered) basis consisting of λ-many open sets. Moreover, we may need our basis to satisfy an additional regularity property: for every R>0, there are at most λ-many basic open set with diameter greater or equal than R. There are a lot of questions arising from this notion. I will mention only a few of them, perhaps the most interesting: are there some good assumptions on λ in order to obtain an analougous of classical properties? Is there a (hopefully, strong) connection between GSTD of Polish Group and Algebraic Topology? How much choice is needed to complete the proofs in the general case?

For instance, we have the following well-known result: every Polish group which admits a left-invariant ultrametric is isomorphic to a closed subgroup of the symmetric group (endowed with the composition of functions). Unfortunately, if we try to generalize the notion of symmetric group, we obtain a completely metrizable group with a weight greater than λ, which is not an element of our theory. Thus, we may consider a proper subgrup of S_λ (called the "bounded symmetric group") consisting of all those bijective functions whose image and preimage of any bounded subset of λ are bounded. With this definition, we manage to obtain a result which is quite similar, but however not "equivalent", to the classical case

References:

• Gao S., Invariant Descriptive Set Theory
• Dimonte, V. Motto Ros, L. Generalized Descriptive Set Theory at Singular Cardinals of Countable Cofinality.
• Becker H., Kechris A.S., The Descriptive Set Theory of Polish Group Actions
• Miller A.W., Descriptive Set Theory and Forcing
• Cramer, S. Implications of very large cardinals