Geometry Seminar

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Counting Realisations for Rigid Graphs

Speaker: Sean Dewar, University of Bristol

Location: Online

Date: Tuesday, February 13, 2024, 2 p.m.

Synopsis:

A graph is d-rigid if for any generic positioning of its vertices in d-dimensional Euclidean space, there are finitely many other realisations of the graph in d-dimensional Euclidean space (modulo isometries) with the same length edges. Combinatorial characterisations for 1-rigidity (i.e., connectivity) and 2-rigidity are known, but it is currently an open problem for d>2. My talk will be a survey of results involving a variation of this problem: given a d-rigid graph with a generic realisation, how many other realisations of the graph exist with the same length edges (now allowing for complex realisations also)? For example, given the 2-rigid graph formed by gluing two triangles at an edge, every generic realisation in the plane has exactly one other edge-length equivalent realisation that is formed by flipping one of the triangles. I will also cover a recent result from Georg Grasegger (RICAM) and I that determines the relationship between the Euclidean and non-Euclidean (e.g., spherical, hyperbolic) variants of the problem.

Notes:

A mix of in person and remote presentations; both types are live-streamed on Zoom and recorded.
In-person and remote talks are at different times.
See mailing list announcements for Zoom details or contact Boris Aronov.