Triple crossing diagrams (TCDs) arise as special cases of planar

bipartite graphs where all the black vertices have degree 3. We define

the notion of a TCD map as an assignment of a point in CP^n to every

white vertex, such that the points at the three neighbors of any black

vertex are aligned. One can associate to any TCD map two collections of

variables that both evolve as cluster X variables under some local

moves. Even more cluster structures appear when considering geometric

operations such as intersecting the lines of a TCD map with a

hyperplane. TCD maps provide a general framework for several geometric

dynamics having a cluster structure. In particular we identify a cluster

structure for Q-nets and several other objects from discrete

differential geometry.

This talk is based on joint projects with Niklas Affolter (TU Berlin),

Max Glick (Google) and Pavlo Pylyavskyy (University of Minnesota).