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Explore the mathematical extension of Wronskian determinants from single-variable functions to multidimensional spaces and their connection to homotopy Lie algebras in this 43-minute conference talk. Learn how the classical Wronskian determinant, used to verify linear independence of N functions in one variable, can be generalized to functions over multidimensional spaces ℝᵈ with Cartesian coordinates. Discover the differential-algebraic identities these multidimensional Wronskian structures satisfy and understand why they follow quadratic, Jacobi-type identities through their relationship to vector field commutation. Examine how the alternated composition of differential operators of strict order p > 0 produces operators with Wronskian determinants as coefficients, leading to the establishment of quadratic identities for higher-order Wronskians. Investigate the connection between these identities and homotopy deformations of Lie algebras, particularly in string theory contexts, and see how multidimensional Wronskians satisfy identities for strongly homotopy Lie algebras. Analyze the dimensional growth problem under iterated N-ary brackets and explore the countable chain of finite-dimensional homotopy Lie algebras that generalize the vector field realization of sl(2) on ℝ. Study the explicit calculation of structure constants and examine the four-dimensional analogue of sl(2) over the plane ℝ² with ternary brackets from Wronskians, representing the only known finite-dimensional homotopy Lie algebra of this type over base dimensions greater than 1.
