Retractions: second order

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Differential Forms Integration

Covers the integration of differential forms on smooth manifolds, including the concepts of closed and exact forms.

Smooth sets and functions: Charts and atlases

Explores smooth functions, n-dimensional charts, compatibility, and atlases for manifolds.

Riemannian metrics and gradients: Why and definition of Riemannian manifolds

Covers Riemannian metrics, gradients, vector fields, and inner products on manifolds.

Covariant derivatives along curves

Explores covariant derivatives along curves and second-order optimality conditions in vector fields and manifolds.

Smooth maps on manifolds and differentials

Covers smooth maps on manifolds, defining functions, tangent spaces, and differentials.

Hands on with Manopt: Optimization on Manifolds

Introduces Manopt, a toolbox for optimization on smooth manifolds with a Riemannian structure, covering cost functions, different types of manifolds, and optimization principles.

Acceleration and geodesics

Explains acceleration along curves and geodesics on manifolds, generalizing straight lines to spheres.

Geodesics in Wasserstein Space

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

Optimization on Manifolds: Context and Applications

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

From embedded to general manifolds: upgrading our foundations

Explores the transition from embedded to general manifolds, upgrading foundational concepts and discussing mathematical reasons for both approaches.