at the

American Institute of Mathematics, San Jose, California

organized by

Tommaso de Fernex and Christopher Hacon

The minimal model program generalizes the classification of surfaces to higher dimensional varieties. After the recent proof of the existence of flips (due to Birkar-Cascini-Hacon-McKernan), one of the main remaining open problems in the field is the Termination of Flips Conjecture. This is an important conjecture with many applications to questions related to the minimal model program. As shown by Shokurov, termination of flips can be reduced to a question on minimal log discrepancies, an invariant that gives a sophisticated measure of singularities. Minimal log discrepancies are known to improve after each flip, and Shokurov conjectured that these invariants have no accumulation points from below, that is, that they satisfy the ascending chain condition (ACC). This conjecture, together with a conjecture on the semicontinuity of these invariants, is known to imply the termination of flips. Results on minimal log discrepancies are of independent interest as these invariants are important in the study of singularities.

The above conjectures constitute the main topics for the workshop:

- Termination of flips.

- ACC for minimal log discrepancies.

These topics are closely related to other central questions in this area of research such as the ACC for log canonical thresholds, the Borisov-Alexeev-Borisov Conjecture, and the Abundance Conjecture. In view of recent spectacular results in this area, we hope that this workshop will help to spur further progress on these conjectures.

The workshop schedule.

A report on the workshop activities.

Papers arising from the workshop:

Ideal-adic semi-continuity of minimal log discrepancies on surfaces

by Masayuki Kawakita, *Michigan Math. J. 62 (2013), no. 2, 443-447 * MR3079272

The index of a threefold canonical singularity

by Masayuki Kawakita, *Amer. J. Math. 137 (2015), no. 1, 271-280 * MR3318092

On log canonical rings

by Osamu Fujino and Yoshinori Gongyo, *Higher dimensional algebraic geometry in honour of Professor Yujiro Kawamata's sixtieth birthday, 159–169, Adv. Stud. Pure Math., 74, Math. Soc. Japan, Tokyo, 2017 * MR3791212

Singularities with the highest Mather minimal log discrepancy

by Shihoko Ishii and Ana Reguera, *Math. Z. 275 (2013), no. 3-4, 1255-1274 * MR3127058

On the log discrepancies in toric Mori contractions

by Valery Alexeev and Alexander Borisov, *Proc. Amer. Math. Soc. 142 (2014), no. 11, 3687-3694 * MR3251710