Abstract for the talk on 13.05.2020 (11:00 h)Seminar on Nonlinear Algebra
Alexander Heaton (Technical University Berlin)
Epsilon local rigidity and numerical algebraic geometry
13.05.2020, 11:00 h, MPI für Mathematik in den Naturwissenschaften Leipzig, G3 10 (Hörsaal)
A well-known combinatorial algorithm can decide generic rigidity in the plane by determining if the graph is of Pollaczek-Geiringer-Laman type. Methods from matroid theory have been used to prove other interesting results, again under the assumption of generic configurations. However, configurations arising in applications may not be generic. We present Theorem 7 and its corresponding Algorithm 1 which decide if a configuration is \(\varepsilon\)-locally rigid, a notion we define. This provides a partial answer to a problem discussed in the 2011 paper of Hauenstein, Sommese, and Wampler. The theorem and algorithm use results from a 2012 paper of Hauenstein. We also present Algorithm 2 which uses numerical algebraic geometry to find nearby valid configurations which are not obtained by rigid motions. When successful, this method demonstrates the failure of local rigidity by explicitly constructing a sequence of configurations which are a discrete-time sample of a continuous flex.