Lectures related to Non-analytic Smooth Functions

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

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

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

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

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

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

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

Explores algebraic topology and differential geometry in understanding robot dynamics and global behavior.

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

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

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

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

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

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

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

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

Covers optimization on manifolds, smoothness, tools needed for optimization, and advanced algorithms using Hessians and Riemannian connections.

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

Explores upgrading foundations from embedded to general manifolds in optimization, discussing smooth sets and tangent vectors.

Discusses linear Lie groups, their definitions, properties, and the relationship between integral curves and vector fields.