Cardinal characteristics above the continuum (ASPEROU17SF) at University of East Anglia on FindAPhD.com

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Dr D Aspero No more applications being accepted Self-Funded PhD Students Only

About the Project

The study of cardinal characteristics of the continuum constitutes a vast area in set theory. A typical question in this area asks whether a given pair of such characteristics can be consistently separated, i.e., it asks a question of the following form: “Is it consistent with the standard axioms for mathematics to have that the smallest set of reals with property P_0 is smaller than the smallest set of reals with property P_1?”. Here, P_0 and P_1 are usually natural combinatorial properties. The method of iterated forcing has provided us with an extremely powerful toolbox for resolving this type of questions and, in the other direction, these questions have historically been one of the main driving forces in the development of iterated forcing techniques. Our understanding of the corresponding characteristics for high analogues of the reals (e.g., the space of all functions from a given uncountable cardinal into itself) is at the moment far more limited. Nevertheless, there are natural questions in this realm that seem to be amenable to recent methods involving iterated forcing with side conditions.

This project aims at developing this area of set theory, focusing both on what can be proved within ZFC, and on the use and development of iteration techniques for the independence direction.

The project may be available at an earlier start date of 1 April or 1 July 2017 but this should be discussed with the primary supervisor in the first instance.

Funding Notes

This PhD project is offered on a self-funding basis. It is open to applicants with funding or those applying to funding sources. Details of tuition fees can be found at http://www.uea.ac.uk/pgresearch/pgrfees.

A bench fee is also payable on top of the tuition fee to cover specialist equipment or laboratory costs required for the research. The amount charged annually will vary considerably depending on the nature of the project and applicants should contact the primary supervisor for further information about the fee associated with the project.

References

i) ) T. Jech, Set Theory, third millennium edition (2003)
ii) K. Kunen, Set theory. An introduction to independence proofs (1980)
iii) D. Aspero, Adding many Baumgartner clubs. To appear in a special volume dedicated to James Baumgartner.
iv) J. Cummings and S. Shelah, Cardinal invariants above the continuum, Annals of Pure and Applied Logic, vol. 75 (1995), 251-268.

Where will I study?

University of East Anglia

We’re one of the UK’s leading research institutions, ranked 10th in the UK for the quality of our research outputs and with more than 1,500 research students.

Cardinal characteristics above the continuum (ASPEROU17SF)

University of East Anglia School of Mathematics