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Explore advanced mathematical concepts in this research lecture examining Nambu-determinant Poisson brackets and their properties on $\mathbb{R}^d$. Delve into the study of Kontsevich's infinitesimal deformations of Poisson brackets using graph complex cocycles, with particular focus on how these deformations preserve the Nambu class. Learn about new nonlinear differential-polynomial identities for Jacobian determinants over affine manifolds, including detailed explanations of the Nambu--Poisson bracket in classical mechanics using the Euler top example. Understand the connection between binary brackets with Jacobian determinant and Casimirs, and their relationship to N-ary multi-linear antisymmetric polyderivational brackets satisfying generalizations of the Lie algebra Jacobi identity. The 53-minute presentation, delivered at Institut des Hautes Etudes Scientifiques (IHES), represents ongoing collaborative research work utilizing the Habrok high-performance computing cluster.
