Differentiating vector fields: Why do it?

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Explores the importance of comparing tangent vectors at different points using algorithms and finite differences.

Delves into topological entropy in compact manifolds and Reeb flows, emphasizing the forcing of entropy via cylindrical contact homology.

Covers second-order retractions in optimization on manifolds, focusing on smooth curves and their relation to the gradient and Hessian of a function.

Explores gradient descent methods for optimizing functions on manifolds, emphasizing small gradient guarantees and global convergence.

Delves into tangent vectors as equivalence classes on manifolds, highlighting their abstract nature and role in optimization.

Explores truncated conjugate gradients for solving the trust-region subproblem in optimization on manifolds efficiently.

Discusses the challenges in differentiating vector fields on submanifolds and the importance of choosing the right method.

Introduces optimization on manifolds, covering classical and modern techniques in the field.

Explores matrix-based and matrix-free approaches for computing the Newton step in optimization on manifolds.

Explores smooth maps, differentials, composition properties, linearity, and extensions on manifolds.

Explores gradient descent with memory, momentum methods, conjugate gradients, and nonlinear CG on manifolds.

Explores geodesics in the Wasserstein space, emphasizing constant speed geodesics and their properties.

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