SINGULAR COTANGENT MODEL

Any singular level of a completely integrable system (c.i.s.) with non-degenerate singularities has a singular affine structure. We shall show how to construct a simple c.i.s. around the level, having the above affine structure. The cotangent bundle of the desingularized level is used to perform the construction, and the c.i.s. obtained looks like the simplest one associated to the affine structure. This method of construction is used to provide several examples of c.i.s. with different kinds of non-degenerate singularities.

The Pentagram Integrals on Inscribed Polygons

The pentagram map is a completely integrable system defined on the moduli space of polygons. The integrals for the system are certain weighted homogeneous polynomials, which come in pairs: $E_1,O_2,E_2,O_2,\dots$ In this paper we prove that $E_k=O_k$ for all $k$, when these integrals are restricted to the space of polygons which are inscribed in a conic section. Our proof is essentially a combinatorial analysis of the integrals.