Pentagram map, complete integrability and cluster manifolds (10rit139)

Objectives

The pentagram map is a projectively invariant iteration on polygons in the projective plane introduced by R. Schwartz in 1992. Extensive experimental and theoretical study lead Schwartz to conjecture and, in part, prove that this dynamical system is completely integrable.

In the spring of 2008, two of us (Ovsienko and Tabachnikov) spent one week in Banff working on symplectic aspect of the pentagram map (the aspect that was missing in the Schwartz investigation). Our stay was very successful: we found an invariant Poisson structure for the map and established its Arnold-Liouville integrability. The results of this research and its follow-up appeared in two papers posted on ArXiv (arXiv:0810.5605 and arXiv:0901.1585; the shorter version was published in Electron. Res. Announc. Math. Sci., 16 (2009), 1--8, and the full version is submitted for publication). The subject has become popular, many researchers (including ourselves) study the relations of the pentagram map with different areas of mathematics.

Our new research project concerns the promising relation of the pentagram map with the so-called "cluster manifolds". Cluster algebras is one of the most interesting and fast growing areas of research of contemporary mathematics. Introduced about 10 years ago by S. Fomin and A. Zelevinsky, cluster algebras are intimately related with representation theory, commutative algebra and symplectic geometry.

Already in 2008, we observed some inexpected resemblance between our formulas and some known formulas in the cluster theory. We discussed this similarity with the leading experts in the field of cluster algebras and tried to understand its origin. We believe that the pentagram map is the first example of an integrable system on a cluster manifold. We hope to find a universal way to generalize this map and to find a similar map for an arbitrary cluster manifold.

It worth mentioning that specialists are trying to find relations of cluster manifolds and cluster algebras with integrable systems. A striking example is a recent proof of the Zamolodchikov conjecture. However, up to now, the well understood integrability dynamical behavior in cluster theory is periodic, whereas the general integrable dynamics is only quasiperiodic. It is the latter that is present in the pentagram dynamics. One of the main concrete aspects of our project is a systematic search for quasiperiodicity in is the Zamolodchikov-like systems.

The third member of our team is Sophie Morier-Genoud (University Paris 6). She is a young expert in representation theory, and her research is closely related to the combinatorics of cluster algebras. She is also actively interested in our work on the pentagram map.