Graph Complex Action on Poisson Structures: From Theory to Computation

Explore the intricate world of Poisson structures and their deformations in this 49-minute lecture by Ricardo Buring at the Institut des Hautes Etudes Scientifiques (IHES). Delve into the theory of Poisson brackets on $\mathbb R^n$, including Nambu–Poisson brackets, quadratic and cubic R-matrix Poisson brackets, and bi-vector fields on $\mathbb R^2$. Discover Kontsevich's infinite family of formulas for infinitesimal deformations of Poisson structures, constructed using graph complex cocycles. Learn about the open problem of finding nontrivial deformations using nonzero graph cohomology classes. Gain insights into the newly developed SageMath software package gcaops (Graph Complex Action on Poisson Structures) and its applications in expanding the class of interesting non-examples. Witness the results of extensive computations, revealing explicit vector fields associated with Poisson coboundaries, complete with concise defining formulas and directed graph representations.