Lectures related to What is a smooth manifold? - Through defining functions

Differential Forms Integration

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

Differential Forms on Manifolds

Introduces differential forms on manifolds, covering tangent bundles and intersection pairings.

Retractions vector fields and tangent bundles: Tangent bundles

Covers retractions, tangent bundles, and embedded submanifolds on manifolds with proofs and examples.

Topology of Riemann Surfaces

Covers the topology of Riemann surfaces, focusing on orientation and orientability.

Optimization on Manifolds

Covers optimization on manifolds, focusing on smooth manifolds and functions, and the process of gradient descent.

Connections: motivation and definition

Explores the definition of connections for smooth vector fields on manifolds.

Smooth maps on manifolds and differentials

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

Riemannian connections

Explores Riemannian connections on manifolds, emphasizing smoothness and compatibility with the metric.

Manifolds: Charts and Compatibility

Covers manifolds, charts, compatibility, and submanifolds with smooth analytic equations.

General Manifolds and Topology

Covers manifolds, topology, smooth maps, and tangent vectors in detail.

Smooth sets and functions: Smooth functions, topology, and manifolds

Explores smooth functions on manifolds, emphasizing continuity and atlas topologies.

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.

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.

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.

Optimality Conditions: First Order

Covers optimality conditions in optimization on manifolds, focusing on global and local minimum points.

Smooth Manifolds: Diffeomorphisms

Explores smooth manifolds through diffeomorphisms and embedded submanifolds in a linear space.

Riemannian metrics and gradients: Examples and Riemannian submanifolds

Explores Riemannian metrics on manifolds and the concept of Riemannian submanifolds in Euclidean spaces.

Manopt: Optimization Toolbox for Manifolds

Introduces Manopt, a toolbox for optimization on manifolds, focusing on solving optimization problems on smooth manifolds using the Matlab version.