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The Erwin Schroedinger Institute for Mathematical Physics

Forcing, large cardinals and descriptive set theory

The field experienced dramatic developments in recent years. A new approach to consistency lower bounds in set theory, which one might refer to as quasi lower bounds, emerged initially from work of Neeman and has been developed further by Friedman-Holy, Sakai and Velickovic and especially Viale-Weiß. Instead of showing that large cardinals are definitively required for set-theoretic properties, one shows that they are required to obtain those properties by the method of forcing, given certain hypotheses on the ground model and the type of forcing used.

A second development is in the application of descriptive set theory to the study of C*-algebras: Farah showed that it is consistent that all automorphisms of the Calkin algebra are inner, and further joint work of his with Toms, Törnquist and others has recently produced dramatic results regarding the unclassifiability of separable C*-algebras. A striking interaction of set theory with ergodic theory is Foreman-Weiss's recent anti-classification theorem for measure-preserving diffeomorphisms of the torus.

In pure descriptive set theory, the exciting ramifications of Ben Miller's recent work reducing numerous dichotomy theorems to variants of the Kechris-Solecki-Todorcevic graph dichotomy are still being worked out. And recently, Friedman-Hyttinen have discovered a connection between higher descriptive set theory and Shelah's classification of first-order theories; many interesting problems remain both on the set-theoretic and model-theoretic sides of this new theory.

The ESI Programme will bring together well established research leaders of the field as well as young postdocs and PhD students to work on these (and other) developments.

 

Schedule

Workshop "Forcing and Large Cardinals", Sep 23 - 27, 2013: PDF

Workshop "Descriptive Set Theory", Sep 30 - Oct 4, 2013: PDF

At a glance

Type: Thematic Programme
When: Sep 09, 2013 to
Oct 18, 2013
Organizers: S. D. Friedman, M. Goldstern, A. Kechris, J. Kellner, W. H. Woodin
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