ESF Research Conferences

Set theory grew out of mathematical analysis through Georg Cantor’s work on sets of uniqueness of trigonometric series in the late 19th century and was soon established as the foundation of all of mathematics. Over the last century it has developed into a vibrant and important subject of its own. On the one hand it deals with questions of mathematical logic of deep foundational importance such as the choice of axioms for mathematics and the questions of relative consistency of mathematical theories. On the other hand techniques of set theory are applied in many areas of mathematics such as classical analysis, general topology, measure theory, Banach space theory, abstract algebra, ergodic theory, and dynamical systems.

The conference will be in honor of Ronald Jensen's lifelong achievements in set theory. Through his work, Jensen has been shaping current day set theory in an outstanding way. His seminal work in the late 60s and in the 70s on the fine structure, the covering property, and the coding capacity of the constructible hierarchy produced a completely new understanding of the set theoretical universe. But it was only a starting point for Jensen's later achievements, which led to the general theory of inner models with large cardinals, their fine structural analysis, the study of their covering properties, and the realisation that the structures of inner model theory are intimately intertwined with hypotheses from descriptive set theory. Nowadays, inner model theory is a prominent research area with its own goals but which also provides other areas of set theory with its forceful tools to bring about deep structural insights that could not have been accomplished otherwise.

The conference focuses on three main topics. The first topic is inner model theory and large cardinals.  While the usual axioms ZFC of set theory are sufficient for most of mathematics, it is well known that they are incomplete and in particular do not decide some of the basic questions of set theory such as the Continuum Hypothesis, its generalisations and variations. Set theorists have been searching for natural extensions of these axioms, which would decide these open problems. There are two basic types of additional axioms which are considered: large cardinal axioms, which postulate that the set theoretic universe is “tall”, and forcing axioms which postulate a certain form of saturation of the set theoretic universe. Both of these directions reinforce Gödel’s basic intuition that additional axioms of set theory should be certain forms of maximality principles.

The second direction is descriptive set theory, which studies properties of definable sets of reals, and more generally Polish spaces. In recent years a number of important developments have brought descriptive set theory closer to ergodic theory, dynamical systems and the theory of group representations. These connections are achieved through the study of orbit equivalence relations and the corresponding quotient spaces. While these spaces are singular, i.e. the Borel structure on them is degenerate, it is possible to study their properties by lifting them to the original space.

Finally, combinatorial set theory deals with uncountable structures without any definability restrictions. This study concentrates on the one hand on independence results, obtained by the methods of forcing and large cardinals, and on the other hand on the complimentary picture of ZFC results obtained through classical combinatorics or by the method of pcf-theory. The interplay of the two forms the intricate nature of the universe of sets, some of whose most important properties have been understood only recently and many of which remain a mystery. Recent progress has solved many long outstanding questions.

The three topics outlined above are naturally intertwined and overlapping. We can list as a fourth scientific topic of the conference applications of set theory to Banach spaces, measure theory, general topology, and other neighboring areas.

Menachem Magidor will give a Special Lecture on the work of Ronald Jensen.

All young researchers are strongly encouraged to bring a poster & participate in the Poster Sessions.