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Explore the latest developments in manifold-based numerical computing through the JuliaManifolds ecosystem in this 31-minute conference talk from JuliaCon Global 2025. Discover new robotics applications including work-in-progress geometric Kalman filters that focus on affine connections, and learn how these mathematical structures enhance computational approaches in robotics. Examine the viability of Manopt.jl as a comprehensive general-purpose optimization framework, featuring extensive solver availability, support for arbitrarily structured decision variables, and impressive performance capabilities. Delve into the ongoing development of statistical tools designed for manifold-valued data, including methods that exploit implicit manifold structures in problems such as robust principal component analysis, and see how these enable rapid prototyping of advanced optimization-based models. Gain insights into future developments including stratification techniques for combining continuous and discrete data, and manifolds with corners for creating more expressive constraint systems. The presentation requires only foundational knowledge of linear algebra, multivariate calculus, and ordinary differential equations, making advanced manifold concepts accessible to a broad audience of computational scientists and engineers.
