About the workshop
Set theory has recently experienced major new developments concerning the construction of models for large cardinals, combinatorial set theory, descriptive set theory, generalisations of Martin's axiom, strong absoluteness and the theory of forcing.
Particularly encouraging is the fact that many unusually talented young people have recently entered the subject, who have established remarkable results concerning the Proper Forcing Axiom and its variants, the Singular Cardinal Hypothesis, subtle properties of stationary sets, and important aspects of applied set theory, including topological dynamics, Borel equivalence relations and metric geometry.
Our two-week workshop was focused around two principal themes: Large Cardinals and Descriptive Set Theory.
Large cardinal axioms play a decisive role in set theory, as they not only resolve many of the questions left unanswered by the standard axioms, as Gödel proposed, but they also provide a way of measuring the consistency strength of virtually any set-theoretical statement. Crucial for the latter is the inner model program of Jensen, which is devoted to the construction and analysis of canonical inner models for large cardinal axioms. Also of interest is the outer model program, which aims to show that large cardinal axioms are consistent with the picture provided by Gödel's universe of constructible sets. Topics to be explored include:
- Inner model theory for a supercompact cardinal.
- The PCF conjecture.
- The consistency of superstrong cardinals with fine structure theory.
- The inner model hypothesis.
- The Ω conjecture.
Descriptive Set Theory
Descriptive set theory is concerned with the study of definable sets of real numbers. Examples of such sets are the Borel sets, and more generally, the projective sets, which are obtained from the Borel sets by taking continuous images and complements. Since all sets of reals that appear naturally in mathematics are projective, their study is of special interest. Descriptive set theory has established many beautiful results regarding Borel equivalence relations and more general Borel relations and also has profound connections with other areas of mathematics, such as ergodic theory, representation theory, harmonic analysis, C*-algebras, infinitary combinatorics and computability theory. Topics to be explored include:
- Borel and analytic equivalence relations and graphs.
- Fraisse limits, topological dynamics and Ramsey theory.
- Countable equivalence relations, measure preserving actions and rigidity theory.
- Theory of turbulence; classification problems in ergodic theory.
- Coding techniques and regularity properties of projective sets.
Funding and Sponsors
Basic funding and orgainsation is provided by the Erwin Schrödinger Instite (ESI). For additional sponsors, follow this link.