Smooth structure

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Optimization on Manifolds: Context and Applications

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

Newton's method on Riemannian manifolds

Covers Newton's method on Riemannian manifolds, focusing on second-order optimality conditions and quadratic convergence.

Tangent Bundles and Vector Fields

Covers smooth maps, vector fields, and retractions on manifolds, emphasizing the importance of smoothly varying curves.

Riemannian metrics and gradients: Why and definition of Riemannian manifolds

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

Smooth Manifolds: Setup

Introduces smooth manifolds, emphasizing the importance of submanifolds of linear spaces.

Gradient Descent

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

Infinitesimal Deformations: One-Dimensional Maps

Explores infinitesimal deformations of one-dimensional maps, discussing common characteristics, methods, and recent results in expanding and piecewise expanding maps.

Embedded Submanifolds: Stiefel Manifold

Covers embedded submanifolds, Stiefel manifold, tangent spaces, and differential ranks.

What is a smooth manifold? - Through defining functions

Explores smooth manifolds through defining functions and submanifolds, highlighting the importance of smoothness.

Total Differential: Definition and Integrals

Explores the definition of total differential and its applications in integral calculus.