Cardinal characteristics above the continuum (ASPEROD_U18SF)
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 much 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 PhD 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 to start earlier than October 2018, but candidates should discuss this with the primary supervisor in the first instance.
This job comes from a partnership with Science Magazine and