![]() ![]() In English, "manifold" refers to spaces with a differentiable or topological structure, while "variety" refers to spaces with an algebraic structure, as in algebraic varieties. The study of manifolds combines many important areas of mathematics: it generalizes concepts such as curves and surfaces as well as ideas from algebra and topology. Riemann’s revolutionary ideas generalized the geometry of surfaces and led to an exact definition of the modern concept of an abstract Riemannian Mannigfaltigkeit, the manifold. His talk ”Uber die Hypothesen, welche der Geometrie zu Grunde liegen” (On the Hypotheses which lie at the Foundations of Geometry) impressed many, including his 77-year-old advisor Carl Friedrich Gauss. Riemannian Geometry: It all started on 10th of June 1854, when Bernhard Riemann (1826 - 1866) gave his famous ”Habilitationsvortrag” in the Colloquium of the Philosophical Faculty at Göttingen. ![]() Welcome to the Fall 2020 course website for Non-Euclidean Methods in Machine Learning (CS468), Stanford University, Department of Computer Science and Geometric Computing Group. I am looking for the skeleton key that opens many doors.” - Andrew Vazsonyi, otherwise known as Weiszfeld I am not looking for hundreds of keys to solve these problems. I want a method that works for lots of problems. “I am just not interested in ad-hoc solutions invented by clever people. Non-Euclidean Methods in Machine Learning “Data has Shape and Shape has Meaning” ![]()
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