Dynamical Algebraic Combinatorics explores maps on sets of discrete combinatorial objects with particular attention to their orbit structure. Interesting counting questions immediately arise: How many orbits are there? What are their sizes? What is the period of the map if it's invertible? Are there any interesting statistics on the objects that are well-behaved under the map?
One particular phenomenon of interest is ``homomesy'', where a statistic on the set of objects has the same average for each orbit of an action. Along with its intrinsic interest as a kind of hidden ``invariant'', homomesy can be used to help understand certain properties of the action. Proofs of homomesy often lead one to develop tools that further our understanding of the underlying dynamics, e.g., by finding an equivariant bijection. These notions can be lifted to higher (piecewise-linear and birational) realms, of which the combinatorial situation is a discrete shadow, and the resulting identities are somewhat surprising. Maps that can be decomposed as products of ``toggling'' involutions are particularly amenable to this line of analysis.
This talk will be a introduction to these ideas, giving a number of examples.