Final Report Summary - SDMODELS (Structured Discrete Models as a basis for studies in Geometry, Numerical Analysis, Topology, and Visualization) The research of the “SDModels” ERC Advanced Grant project connected traditionally quite distant fields of current mathematical research via common or structurally similar discrete (mostly: geometric) models. It made substantial contributions to mathematical research by highlighting, developing, and exploiting theory for common structures and structural similarities that occur in problems/theories from diverse mathematical application areas.Thus research in Focus Area I: High-Complexity Geometry of the project has thus produced surprising new insights into the connection between rigidity and symmetry, and it has developed construction methods for polyhedral structures where the pattern of vertices, edges and faces (the “combinatorics”) alone determines the coordinates of the structure uniquely – up to a projective transformation. It was thought for a long time, since the studies of Shephard and Perles in the 1960s, that in each dimension there could be only finitely many polyhedra that are “projectively unique” in this sense. By combining methods from rigidity with intricate geometric constructions, Adiprasito and Ziegler proved that this is completely wrong for polyhedra in a very high-dimensional space. In Focus Area II: Delaunay Geometry the SDModels research concerned geometric patterns defined by circles and spheres. This gives completely new insights into a problem first posed in 1832 by the famous geometer Jakob Steiner: Can all combinatorial types of polyhedra be realized such that all vertices lie on a sphere? The answer has long been known to be “no”, but a complete characterization of the types of polyhedra that have such a representation remains elusive – in particular if we also consider polyhedra of higher dimension. Gonska and Ziegler have now solved the problem for the case of so-called “stacked” polyhedra, which can be obtained by glueing simplices (the higher-dimensional analogues of tetrahedra) one after the other into a starting simplex.In Focus Area III: Topological connectivity the point of view is more global – one looks, for example, at the space of all realizations of a combinatorial structure, or for coincidences that must always occur when complicated and highly-connected structures are drawn in the plane.For example, if all connection lines are to be drawn between five distinct points in the plane, then two of them must cross. “Tverberg type problems” consider larger numbers of points, and then conclude that, for example, if all connection lines are drawn between seven points, then either one of the points is contained in two triangles spanned by two other groups of three points, or an intersection point between two lines lies in a triangle spanned by the remaining three points. The SDModels project has produced a strikingly simple and powerful method, which they called the “constraint method” which can be used to prove Tverberg-type results, such as the “Colored Tverberg Theorem” – but it finally also has been used to disprove the general, “topological” Tverberg problem, a celebrated success achieved by Florian Frick together with Pavle Blagojevic and Günter M. Ziegler in the framework of the SDModels project (based on work by Isaac Mabillard and Uli Wagner in Vienna).
