The pentagram map is a projectively natural transformation defined on (twisted) polygons. A twisted polygon is a map from Z into RP 2 that is periodic modulo a projective transformation called the monodromy. We find a Poisson structure on the space of twisted polygons and show that the pentagram map relative to this Poisson structure is completely integrable. For certain families of twisted polygons, such as those we call universally convex, we translate the integrability into a statement about the quasi-periodic motion for the dynamics of the pentagram map. We also explain how the pentagram map, in the continuous limit, corresponds to the classical Boussinesq equation. The Poisson structure we attach to the pentagram map is a discrete version of the first Poisson structure associated with the Boussinesq equation. A research announcement of this work appeared in [16].

We prove the existence and the uniqueness of a conformally equivariant symbol calculus and quantization on any conformally flat pseudo-Riemannian manifold (M, g). In other words, we establish a canonical isomorphism between the spaces of polynomials on T * M and of differential operators on tensor densities over M , both viewed as modules over the Lie algebra o(p + 1, q + 1) where p + q = dim(M ). This quantization exists for generic values of the weights of the tensor densities and compute the critical values of the weights yielding obstructions to the existence of such an isomorphism. In the particular case of half-densities, we obtain a conformally invariant star-product.

We introduce a notion of q-deformed rational numbers and q-deformed continued fractions. A q-deformed rational is encoded by a triangulation of a polygon and can be computed recursively. The recursive formula is analogous to the q-deformed Pascal identitiy for the Gaussian binomial coefficients, but the Pascal triangle is replaced by the Farey graph. The coefficients of the polynomials defining the q-rational count quiver subrepresentations of the maximal indecomposable representation of the graph dual to the triangulation. Several other properties, such as total positivity properties, q-deformation of the Farey graph, matrix presentations and q-continuants are given, as well as a relation to the Jones polynomial of rational knots.