Apparaît dans la collection : The Pentagram Map and Ramifications

In the definition of the pentagram map, both space and time variables are discrete, but play unequal roles. Unifying time and space leads to the notion of pentagram nets. A pentagram net is an assignment of points to the edges of a regular triangular lattice such that the points assigned to the edges of every elementary lattice rhombus are collinear. We show that pentagram nets are critical points of a simple area-like functional, yielding a variational description of the pentagram map and, consequently, an invariant presymplectic structure. In contrast to the known Poisson bracket for the pentagram map, arising from the trigonometric r-matrix, the new structure is related to the rational r-matrix. Moreover, we show that pentagram nets are integrable in a natural sense. The existence of integrals in this setting is a consequence of a high-school-level theorem about triangles. Restricting to the pentagram map, we recover the known integrals. Together with our new presymplectic structure, this yields integrability of the pentagram map directly on the space of planar polygons, without passing to twisted polygons or taking a quotient by the projective group. Time permitting, we may also discuss multidimensional pentagram nets and how they lead to dented pentagram maps, Grassmannian pentagram maps, and some new integrable pentagram maps; connections to Glick’s results on the limit point of the pentagram map; connections to Veselov’s work on solitons for matrix KdV; and connections to the Adler–Bobenko–Suris theory of 3D consistency.