Christopher Hacon is the McMinn Presidential Endowed Chair and Distinguished Professor at the University of Utah. He is a Fellow of the Royal Society, a member of the National Academy of Sciences and of the American Academy of Arts and Sciences. His research in algebraic geometry has been recognized by the Clay Research Prize 2007, the Cole Prize 2009, the Feltrinelli Prize 2011, and the Breakthrough Prize in Mathematics 2018. Chenyang Xu is Professor at Princeton University. He is a fellow of the American Mathematical Society. His contributions to algebraic geometry have been recognized by the Future Science Prize 2017, the New Horizon Prize 2019, and the Cole Prize 2021.

• 1. Foreword; 2. Birational geometry of algebraic varieties and Shokurov's work; 3. ACC for log canonical thresholds for complex analytic spaces; 4. Conjectures on the Kodaira dimension; 5. Characterizing terminal Fano three folds with the smallest anti-canonical volume; 6. Uniform rational polytopes for Iitaka dimensions; 7. MMP for algebraically integrable foliations; 8. On Toric Fano fibrations; 9. Q-Fano three folds of Fano index thirteen; 10. Reflective 2-elementary lattices; 11. The relative duBois complex-on a question of s. Zucker; 12. Factorization presentations; 13. Spectrum bounds in geometry; 14. On the DCC of Iitaka volumes; 15. Shokurov's index conjecture for quotient singularities; 16. A note on the Sarkisov program; 17. Cluster varieties and Toric specializations of Fano varieties; 18. Birational rigidity and alpha invariants of Fano varieties; 19. On f-pure inversion of adjunction; 20. On termination and fundamental groups; 21. Motivic integration on Berkovich spaces.

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