James Cummings is a Professor of Mathematical Sciences at Carnegie Mellon University in Pittsburgh, Pennsylvania. Ernest Schimmerling is a Professor of Mathematical Sciences at Carnegie Mellon University in Pittsburgh, Pennsylvania.

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• Contributors; Introduction; 1. An introduction to Pmax forcing Paul B. Larson, Peter Lumsdaine and Yimu Yin; 2. Countable Borel equivalence relations Simon Thomas and Scott Schneider; 3. Set theory and operator algebras Ilijas Farah and Eric Wofsey; 4. Set mapping reflection Justin Moore and David Milovich; 5. An introduction to hyperlinear and sofic groups Vladimir G. Pestov and Aleksandra Kwiatkowska; 6. Aronszajn trees and the SCH Itay Neeman and Spencer Unger; 7. Iterated forcing and the continuum hypothesis Todd Eisworth, Justin Tatch Moore and David Milovich; 8. Short extender forcing Moti Gitik and Spencer Unger; 9. The complexity of classification problems in ergodic theory Alexander S. Kechris and Robin D. Tucker-Drob; 10. On the strengths and weaknesses of weak squares Menachem Magidor and Chris Lambie-Hanson; 11. Proper forcing remastered Boban Veličković and Giorgio Venturi; 12. Set theory and von Neumann algebras Asger Törnquist and Martino Lupini; 13. The HOD dichotomy W. Hugh Woodin, Jacob Davis and Daniel Rodríguez.