at the

American Institute of Mathematics, San Jose, California

organized by

James Propp, Tom Roby, Jessica Striker, and Nathan Williams

- promotion and evacuation for Young tableaux;
- the action of a Coxeter element on a parabolic quotient of a Coxeter group; and
- crystal operators on highest-weight representations.

- promotion and rowmotion for order ideals and antichains in posets; and
- their piecewise-linear and birational liftings.

Some of the properties of these cyclic actions can be explained by the importation of combinatorial or algebraic models that explain why the action exists. When the cyclic action has predictable orbit structure, this program has been very successful (as seen in the recent flurry of work on the cyclic sieving phenomenon). The encoding of alternating sign matrices under gyration by fully packed loops and their associated link-patterns shows that such models can exist even when the orbits of the cyclic action display resonance and some are quite large. We hope to study further actions of this last sort, such as rowmotion on plane partitions of height greater than two.

Some examples of problems we are interested in are:

- Develop a combinatorial model of alternating sign matrices of size n that explains the existence of the cyclic action, superpromotion, of pseudo-period $3n-2$ with properties similar to gyration.
- Uniformly prove that birational promotion and rowmotion have finite order on all minuscule posets.
- Express known combinatorial actions as compositions of piecewise-linear involutions and investigate their birational analogues.
- Uniformly prove a bijection between nonnesting partitions and clusters related to Panyushev's homomesy conjectures.

- To produce new combinatorial models that explain the existence of known cyclic actions and homomesies.
- To use data provided by cyclic actions, invariants, and homomesies to produce new bijections between combinatorial objects.
- To coordinate work on homomesy and generalized toggle group actions.
- To suggest directions for future research.

The workshop schedule.

A report on the workshop activities.

A list of workshop notes and open problems, by Sam Hopkins.

A pre-workshop problem list prepared by the organizers.

One of the outcomes of the workshop was the creation of a Dynamical Algebraic Combinatorics ("DAC") listserv. To join, send email to James Propp.

Papers arising from the workshop:

Resonance in orbits of plane partitions and increasing tableaux

by Kevin Dilks, Oliver Pechenik and Jessica Striker, *J. Combin. Theory Ser. A 148 (2017), 244–274 * MR3603321

Noncrossing partitions, toggles, and homomesies

by David Einstein, Miriam Farber, Emily Gunawan, Michael Joseph, Matthew Macauley, James Propp and Simon Rubinstein-Salzedo, *Electron. J. Combin. 23 (2016), no. 3, Paper 3.52, 26 pp * MR3576300

The CDE property for minuscule lattices

by Sam Hopkins, *J. Combin. Theory Ser. A 152 (2017), 45–103 * MR3682727

Braid moves in commutation classes of the symmetric group

by Anne Schilling, Nicolas M. Thiéry, Graham White and Nathan Williams, *European J. Combin. 62 (2017), 15–34 * MR3621720

Genera of Brill-Noether curves and staircase paths in Young tableaux

by Melody Chan, Alberto López Mart&\acute;in, Nathan Pflueger and Montserrat Teixidor i Bigas, *Trans. Amer. Math. Soc. 370 (2018), no. 5, 3405–3439 * MR3766853

The expected jaggedness of order ideals

by Melody Chan, Shahrzad Haddadan, Sam Hopkins and Luca Moci, *Forum Math. Sigma 5 (2017), e9, 27 pp * MR3623577

Poset edge densities, nearly reduced words, and barely set-valued tableaux

by Victor Reiner, Bridget Eileen Tenner and Alexander Yong, *J. Combin. Theory Ser. A 158 (2018), 66–125 * MR3800124

Noncrossing partitions, toggles, and homomesies

by David Einstein, Miriam Farber, Emily Gunawan, Michael Joseph, Matthew Macauley, James Propp, Simon Rubinstein-Salzedo