## Projektdetails

### Abstract

Wider Research Context

The project aims to study of the combinatorics and definability properties of entire functions. This topic originates in work of Erdos from the sixties and was further developed in the subsequent decades, more recently by Burke, Kumar and Shelah.

The starting point is Erdos’ result that the continuum hypothesis holds if and only if there is a family F of entire functions such that for every complex number z the set of values f(z) is countable. He asked whether or not it is consistent that the continuum hypothesis fails while for every complex number z the set of values f(z) has fewer elements than the continuum. Recently, Kumar and Shelah answered this question affirmatively. Curiously, however, in the model they provide, the continuum is of a singular cardinality and so the status of this problem remains open in models of forcing axioms in most of which the continuum has the third-smallest infinite cardinality.

Approach:

Consistency results would be proved by iterated forcing. First one would attempt iterating proper forcings with countable support as this allows for a comparatively wide range of iterands. This approach, however, necessarily results in a model where the continuum has the third-smallest infinite cardinality. To prove that a statement is mutually consistent with the continuum being larger than that, one would typically attempt to iterate ccc forcings with finite support. This approach has been developed further in recent years, among others by the co-applicant.

A technique which is generally useful in infinite combinatorics and most probably will be of use here regardless of what the answers to the problems turn out to be is that of transfinite induction.

The definability properties would be analysed by adapting a method pioneered by A. Miller which has been adapted to quite a few different settings but not yet to complex analysis.

Innovation

Within set theory the aims of the project are somewhat analogous to existing lines of research but tackling a problem which shows some odd features such as up to now only yielding to analysis when the cofinality of the continuum is the smallest possible. Therefore it is very likely to provoke innovation within infinite combinatorics. Another aim is to bring set theory closer to the rest of mathematics and classical complex analysis with its long history and wide scope probably is one of the most attractive potential companion subjects within this rest.

Primary Researchers Involved

Beside the applicant and the Co-applicant, the project would involve Ashutosh Kumar from the Indian Institute of Technology in Kanpur who was a postdoctoral colleague of the applicant in Jerusalem in 2014 and 2015 and whom he is collaborating with presently. He worked together with Saharon Shelah on the topic of the project. It would moreover involve Maxim Burke from the University of Prince Edward Island who has credentials relevant to the project in the theory of entire functions.

The project aims to study of the combinatorics and definability properties of entire functions. This topic originates in work of Erdos from the sixties and was further developed in the subsequent decades, more recently by Burke, Kumar and Shelah.

The starting point is Erdos’ result that the continuum hypothesis holds if and only if there is a family F of entire functions such that for every complex number z the set of values f(z) is countable. He asked whether or not it is consistent that the continuum hypothesis fails while for every complex number z the set of values f(z) has fewer elements than the continuum. Recently, Kumar and Shelah answered this question affirmatively. Curiously, however, in the model they provide, the continuum is of a singular cardinality and so the status of this problem remains open in models of forcing axioms in most of which the continuum has the third-smallest infinite cardinality.

Approach:

Consistency results would be proved by iterated forcing. First one would attempt iterating proper forcings with countable support as this allows for a comparatively wide range of iterands. This approach, however, necessarily results in a model where the continuum has the third-smallest infinite cardinality. To prove that a statement is mutually consistent with the continuum being larger than that, one would typically attempt to iterate ccc forcings with finite support. This approach has been developed further in recent years, among others by the co-applicant.

A technique which is generally useful in infinite combinatorics and most probably will be of use here regardless of what the answers to the problems turn out to be is that of transfinite induction.

The definability properties would be analysed by adapting a method pioneered by A. Miller which has been adapted to quite a few different settings but not yet to complex analysis.

Innovation

Within set theory the aims of the project are somewhat analogous to existing lines of research but tackling a problem which shows some odd features such as up to now only yielding to analysis when the cofinality of the continuum is the smallest possible. Therefore it is very likely to provoke innovation within infinite combinatorics. Another aim is to bring set theory closer to the rest of mathematics and classical complex analysis with its long history and wide scope probably is one of the most attractive potential companion subjects within this rest.

Primary Researchers Involved

Beside the applicant and the Co-applicant, the project would involve Ashutosh Kumar from the Indian Institute of Technology in Kanpur who was a postdoctoral colleague of the applicant in Jerusalem in 2014 and 2015 and whom he is collaborating with presently. He worked together with Saharon Shelah on the topic of the project. It would moreover involve Maxim Burke from the University of Prince Edward Island who has credentials relevant to the project in the theory of entire functions.

Status | Nicht begonnen |
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### Projektbeteiligte

- Universität Wien (Leitung)
- Indian Institute of Technology Kanpur
- University of Prince Edward Island

### Schlagwörter

- entire
- forcing
- continuum
- function
- cardinal
- family