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Explore cutting-edge developments in diffusion geometry through this 51-minute research talk that introduces a novel framework for geometric and topological data analysis. Discover how Riemannian geometry can be defined for probability spaces, enabling the application of classical differential geometry theories to modern data analysis. Learn about fundamental concepts including vector fields and differential forms construction, while examining practical applications such as vector calculus, spatial PDE solutions on datasets, integral curves, geodesics computation, and circular coordinate determination for de Rham cohomology classes. Gain insights into specialized applications for manifold data, including curvature tensor calculations and dimensionality analysis, with emphasis on the framework's advantages in noise resistance and computational efficiency compared to traditional methods like persistent homology.
