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Explore recent developments in integral equation methods through this one-hour conference talk that focuses on two critical problem classes: systems of two-point boundary value problems and parabolic partial differential equations in moving geometry. Learn how the first class applies to diverse computational challenges from dynamical systems to optimal control problems, while the second proves essential for numerical simulations in crystal growth, diffusion MRI, and complex fluid dynamics. Discover how selecting appropriate integral formulations and incorporating advanced fast algorithms—including fast direct solvers and the newly developed adaptive fast Gauss transform—creates numerical methods that achieve speed, adaptability, and high accuracy with arbitrary order convergence. Examine numerous numerical examples that demonstrate the practical effectiveness of these approaches across various scientific and engineering applications.
